Everybody forced here ripping three boxes of 21 22 upper-deck black diamond CDD exclusive hockey boxes here Three of them three of them Patrick T Patrick. Good luck Let's see what you got CDD exclusive Love black diamonds one of the nicest hockey products out there One feels a little thicker. I don't I can't remember. I can't remember. I only open one so far. So starting off Matthew could chuck 295 out of 349 That's right stick puck time And now they're just getting all the false starts and everything now they're just backing up again This is sick Black diamond band of color rookie Anton Lindell six of 22 These are stunning That is fantastic. That's just a cool card That is really cool. Yeah, I I agree von that is a sexy card Band of color love that Six diamonds piece. Yeah, that's awesome. I dig that. That's cool Never seen something like that before Got a cup winners patch mario la mu manufactured patch Are the greatest player ever Just got her just got her. I didn't notice the band of color. There were diamonds as well Yeah diamonds. Oh, I like this one a lot triple patch patrice bergeron brad marshan and david posternock Love that one Yep, all game use sweet Bruins triple there diamond mine triple relics bergeron marshan and pasta That's an awesome one not on the same line anymore though Rookie gems Anton Lindell 129 of 399 Oh, nice exquisite collection Exquisite moments seattle kraken July 21st 2021 to host expansion drab 239 of 399. That's a cool card Anything exquisite is awesome. And on the back is rookie gem spencer knight Nice night 30 of 99 The big rookie auto this time There we go. All right darnel nurse ones 65 out of 349 anders lee black diamond auto sparkling scripts Anders lee not numbered rookie team logo jumbo sheen pinto senators jumbo patch there manufactured These are so cool looking It's like a die cut with acetate behind a diamond gallery quinton byfield I don't like holding these though because you can't hold them on like the sides Where I normally hold them in the center Nice byfield be very careful Yeah, just leaving that one. Maybe just I don't know cool looking byfield rookie gems joe valeno 362 of 399 exquisite collection rookie patch ross coulton 15 of 25 Love the exquisites player worn on the patch And rookie gem is joe valeno Number 52 and 99. Yeah, we're gonna mag that byfield not the biggest card, but it just Doesn't seem safe to sleeve it when when you get up to that second Yeah, exactly really hard. All right last one. Let's see the heat. Let's see the heat Last box in the building got a special to cove 66 at 349 Andres special to cove Got a hendrix lapierre 25 of 49 diamond relic rookies Very nice dual looks like a ruby gem like a dual red diamond hendrix lapierre Nice card there team jumbo logos mckelgulay A um a kebek nord deeks On the manufactured patch The nord deeks bring them back bring them back Better than the current Team in kebek Just kidding guys, but am I Conor hella buck diamond mine relic Buck not numbered diamond stars on ze copetar 12 of 249 rookie exquisite Is cam york 86 of 99 first round pick for the flyers in 2019 And rookie gems is alex barry bullay 84 of 99 There we go. Thanks again patrick Appreciate it man. Sorry couldn't pull another z grass like it hit yesterday, but that was an awesome card Appreciate it. We'll get those right out to you


OK, Marie, you played against Hussein today and the game ended in a draw. Tell us something about the game. I was not very good in my game. I didn't get very good position and I don't know what to do. I would like to attack on the king sign but it was very solid. So you agreed to draw just after the time control, yes? Who do you think was better in the game generally? Was it you or Hussein? It was a very solid game, very complicated game. There are chances for the boss but little chance. Fairly level game. OK, let's have a look at it. OK, Marie, we're looking at a position from your game where you've got a queen on H4 and a possibility of taking the H-port. Yeah, but in the game it's very difficult to do this because I saw the variance. I now bishop H2 and my queen. Your king is in some danger. Yeah, but I don't know, G5 and G6.  Yeah, but maybe it's solid because I have one pawn more. G5 is not very good, maybe G6 instead of G6. G5 and... So do you think he would escape and come through with a pawn up in that position? I don't know, but it was too risky. It was too risky. Yeah. I tried to come up with G6, queen of S5, G5, I like for black. Yeah, yeah, so... I like to play this with black, not with white. Right. OK, so you... I'm under the pressure and he can attack on the caster, the king.  Yeah, I'd be proud to attack him. And are concentrated on putting pressure on the queen's side. I know this, but now I change all the pieces. OK. And if I attack like this with a queen, I can just trap my queen. And take the pawn, yeah. Attack like this on my rookies. So what's the assessment about that position? Now it's really... Oh, I'm sorry, you're trapped here? Yeah, yeah, I'm trapped. My king, my rook is trapped. It's very nice. OK. But let's leave the pawn. Yeah, so you couldn't... So you have to take with a rook in that position. And this, and I can... Queen F2, I want to push. You want to push F4, yeah? Yeah. Yeah. Take. And take on c5. Yeah, this and now F4. And now it looks mid-robe because I have to change all the pieces. Yeah. Yeah. Right. So the game just peters out to a draw from here, does it? Nothing interesting happens after this? I think no, just... And after, it's like I try to attack something. Right, you go back to attacking the H pawn. Maybe this, it can be interesting. It could be attacks. Queen g5, yeah. Yeah. But you played queen g4, yeah? Yeah, queen g4 instead of this. And then there's big exchanges. Rook a2 threatening the c4. And now it looks very good form, like it's okay. Yeah, it's very solid. Yeah, take, take, and just protect. Yeah. And I cannot go for the... Because it comes with b5 and it comes with king. Right. King invades through, comes through here or something. Yeah. Okay. Okay, thank you very much.


Don't like that.  I'm Rick. You can follow the Instagram, Twitter for more juicy content Thanks to Patreon, post, subscribe, like, button. I'm like 30 years old Never once paid my taxes Like Donald Trump, exactly Anyways today, we got a video. This is a ten Bollywood actors who refused famous movie roles. Ah Refused them. I guess Yeah, and that happens all the time. Yes, you know the what you know the big one about Avatar, right? It was Brad Pitt, right? No, it was Matt Damon Matt Damon, which Matt Damon turned it down and and What about her he he he Cameron told him he was gonna give him 2% of the box off Was a 10 I think it was 10 whatever it was it was a lot of a lot of money On top of all the other movies that yeah But yeah, there's a lot of famous people who had been offered parts and they didn't take them and then someone got it and they Turned it into part of the business role. Yeah, because whether it's they didn't want it or a lot of time They couldn't do it. Yeah, they just couldn't do it They probably want to do it really bad and they just they were doing a different film at the time or What's going on or you have a really beautiful story like the original person to play Lincoln who was signed at the time was Liam Neeson and It just kept getting pushed back schedule conflicts things of different nature and the person Spielberg Originally approached for it was Daniel Day-Lewis and at the time Daniel Day-Lewis felt he was too young to play Lincoln time goes on Daniel Day-Lewis becomes available and it's Liam Neeson who says to Steven Spielberg You might want to revisit that because if you can get Daniel Day-Lewis you should yeah And he just gave the roll up because he felt that it should go to Daniel Day-Lewis. That's that's an incredible person Most roles should yeah, here we go See in any kind of industry you have to be at the right place at the right time Yeah, a lot of people account the initial success to that be the young Alia Bhatt auditioning for student of the year Also, Shaan Singh Rajput and the many years he spent on Indian television But there are some cases when actors are fortunate to get roles because of some refusal and those roles that they eventually take up Become their most iconic and memorable ones So here is tried and refused productions with top 10 Bollywood actors who turned down famous roles Kangana Ranawath After her success in movies such as fashion and once upon a time in Mumbai Kangana was offered the iconic lead role in the dirty picture Considered to be half-breaking cinema and a movie that changed the way She probably would have done really well trade on screen the movie later was offered to Vidya Baland And went on to come her really job performance in her career Kangana recalling the time she was offered the role in an interview with Rajiv Masan Told everyone why she turned down the role in many similarities about a small town girl wanting to make it big and Then falls a prey to that whole glamour and that façade I just felt I'll be trapped in this whole thing doing the same that makes sense Yeah, and I was bored of seeing myself. I like her eye makeup I'm a very attractive person and then crying and sobbing my way to my deathbed Akshay Kumar The actor who at the time was primarily doing action comedies which required little to no effort Movies including Desi boys houseful to and Joker was offered to portray the iconic role of milk I sing in Bagh milka. Oh, wow The actor turned down the role for unknown reasons and it's surprising to know that even milk I sing picture Akshay Kumar to play him in the movie for an actor went on to play the actor who went to old And definitely put forth an award winning what year came out one indeed Rittik Roshan Rittik Roshan before doing coin milga and Laksh in the early 2000s was doing a huge set of embarrassing romantic movies like yadee na tum jano na hum and Aap mujhe achhe lagne lage the actor in this period was offered for this and turned down the role Shahrukh went on to take the role of Mohan Bargav, which to date is one of his most underrated Performed you probably would have done well. He would have done well on that as well Sharma Khan did really well in the box office Salman Khan the 2007 sports film which focused on Fresh change in Indian cinema proving that scripts and stories focusing on relevant topics can also become a box office success Shahrukh Khan who played the memorable role of anymore unless your KGF fourth an amazing performance But little known trivia is that Salman Khan was actually offered the role first Which he of course turned down in a DNA interview Salman jokingly said Chuck they was offered to me But I left it because even Shahrukh Khan should be a part of a few great films That's funny that's funny Great Are the in fact told me that you know that I will make this into like the biggest hit and you don't know what you've been missing out on but I didn't think You know because I was I was fortunate that partner did well as well and People have loved the film Rittik Roshan and Abhishek Bachchan The 2001 super hit Dil Chhata is in my opinion Farhan Akhtar's finest directorial work And it's extremely interesting to know the initial casting process for the movie Farhan Akhtar has been a film family kid and grew up with Rittik Roshan and Abhishek Bachchan And he wanted to cast his childhood friends in his first movie The entire fiasco of setting a final cast is vividly recalled by Farhan Akhtar in a TV interview with and picture So my first choices actually Akshay Khanna Rittik or Abhishek, they were my first three choices Only person who I got to do the film was was Akshay Shahrukh Khan Shahrukh Khan in the late 90s and early 2000s was doing movies such as K3G, Dev Das and Saathya And had definitely reached the A list celebrity status during this time Shahrukh Khan was offered Rajkumar Hirani's director Wow The actor refused the iconic role Rumored to be due to his back problems and had multiple meetings with Rajkumar Hirani The movie is working with him now Donkey Rajkumar Hirani even before approaching producers in SRK had Anil Kapoor in mindful role But Sanjay Dutt took the role as Munna Which ultimately gave him an image makeover and helped change the public perception of the controversial superstar This entire ordeal is discussed in an interview with Anupama Chopra by the director From Vivek to Shahrukh, when it came to Shahrukh, I just couldn't Believe that that's a possibility. That's a possibility. I thought that either Shahrukh or I will do Finally, of course, long journey and your end credits say thank you to Shahrukh Khan We had many meetings and Yeah, a lot of fun we had but Because this time Sanjay was this time to be in the film and Saif Ali Khan Everyone remembers Saif Ali Khan for his roles in Omkara and Salam Namaste But in 1995 Saif wasn't an established star doing movies like Intiha and Meh Khiladi, Tuva Nadi Did you know that when Aditya Chopra was casting for his directorial debut in 1995 Saif Ali Khan could have been cast as a Raj in DDLJ Aditya approached Shahrukh Khan to play the role of Raj He was initially not interested because of the romantic nature of the role having had success in action movies like Bhajigarh and Dhar Aditya then asked Saif Ali Khan to play the lead role Wow That would have been different declined for unknown reasons Wow Causing Aditya to continue pursuing Shahrukh Khan Good thing he did Four meetings over several weeks Wow That's the biggest bomb show right there That is astonishing Karina Kapoor The actress can be considered to be a career maker in the film industry as she has refused several roles that have gone on to become actresses Most iconic and best performances The three-dead star after production and creative issues walked out of Kahona Pyaa Hai Later she said no to Hum Dil Decheke Sanam Which pair headed Aishwarya's career in the industry The actress also went on to say no to Kal Hona Ho because of a friendship fallout between her and Karan Johar Discussed in the first season of coffee with Karan The movie went on to become one of the biggest hits and one of Preeti Zinta's best performances Even further Karina had initially said yes to the 2013 smash hit To star opposite Ranveer Singh But after creative differences and major payment and scheduled problems walked out of the film Leading to quite the war of words It wouldn't have been It wouldn't have been Ranveer The actress could have had a plethora of smash hits if you think about it Shah Rukh Khan Shah Rukh Khan makes it again on our list The Oscar nominated movie Laganh which literally got all forms of acclaim is definitely one of the best Bollywood movies in the 20s Some of these are mind boggling That would have made it very different Laganh went on to do wonders for Indian cinema on an international level Little known fact is that Ashutosh Gowaragar actually approached Shah Rukh Khan for the role first In an interview with Rediff Ashutosh talked about casting SRK See bovans definitions change as does the script. It's like a paradigm shift Every actor imparts his personality to a character the moment he does something the character undergoes a bit of change And when he does it everything around him changes So obviously Shah Rukh would have played a different absolutely discuss the number one position here are some honorable mentions Interesting Wow actually started shooting Oh, wow Sanjeev The movie that changed the trend from romantic films to action films and pioneered Amitabh's new image of the angry young man Sanjeev was an iconic movie that started a new wave in Hindi cinema This film also ended the struggling period of Amitabh and turned him into a rising star A movie that did so much for the actor was surprisingly offered to several actors before him The role of the lead inspector Vijay Khanna was first offered to Dharmendra Who agreed but his brother Ajith who was co-producing the film fell out with Prakash Mehra and the actor backed off Dilip Kumar was offered the movie after that, but he thought the role was too one-dimensional Dev Anand was next in line But the creators after discussions concluded he was a misfit for the role Raj Kumar came on later who had already reprised the role of a cop in Unche Logue Rejected the film and finally the struggling actor Amitabh Bachran After a lot of debate and creative meetings got the film and everything changed post the movie for people A lot of people say that's one of the big B ones we need to get to Yeah, I think we have it Amitabh Bachran in an interview with Rajeev Masand recalls the reason why he got the role in Zanjeer Prakash Ji had Come to me after this film had been recommended to several other Senior artists for some reason Jagat Saab had noticed an action sequence of mine in a film called Bombay to Coa Where I was chewing up and in the middle of an action sequence. I got hit in the face I rolled over stood up again and apparently I was still chewing up And he felt that after I saw that moment I decided that you would be able to do this role and that was the list guys Wow, that was good. There's some big bomb shows in there that blew my mind And obviously I'm sure if you look at Hollywood Wise as well. There's massive roles. I mean Johnny Depp was almost fired from pirates multiple times because they didn't understand what he was doing Yeah, the first few days of I was telling that to Indrani the other day when we went to Disneyland for the first time and she went into pirates I told her that Johnny has shown up and surprised people And just the choices that were made up when the daily's the producers saw the dailies for the first few days Their response was what the hell is he doing? He's going to ruin this movie. We've got to find somebody else to play captain jack Yeah, and there's probably 10,000 stories like that about 10,000 different actors It's just you know who was going to be indiana jones who it was offered to first. I think I've heard it before but I can't remember who Tom Selleck Interesting. Yeah, it was going to be Tom Selleck instead of Harrison Ford. Yeah And just like all these if you put a different actor in it A lot of times and there's some that it's like, oh, yeah, I could see that they'd probably do something a little similar But it completely changes it of course because the actors always brave themselves Shahrukh Khan is in laghan. It's going to be completely different than Amir Khan in laghan It may still be good, but it's going to be completely different and it it's not necessarily going to work because it doesn't only what they said it doesn't only Uh, it embody the character with something new because that person is in it But all the other parts that are played are going to be slightly changed because they're responding to what that Person's bringing. Yeah. Yeah. Yeah, really good video. Very fun to learn all this, uh, old school stuff about, uh, The history of mind-blowing ones in there. So if there's other videos we can react to please let us know what those are down below


My name is R&B, I'm going to be doing the Whiskey 6, Claimor Mine Lane, and I'm going to go ahead and walk myself through the steps as I'm performing the lane. So the first thing I want to do is go ahead and take the prone, and I'm going to go ahead and get my band of leader right here on my work space. Then my tom will start, three, two, one. So I'm going to go ahead and reach in for my tester, which is my M57 initiator or clacker, and then I'm going to go ahead and get my tester M40 tester. I'm going to go ahead and connect them both. Once they're connected, I'm going to go ahead and put it on fire, by depressing the right side, the springy side, on my clacker, and opening that up, and then I'm going to couple with my free hand, I'll put it right next to my face, and depress the clacker. I'm going to stay, I see the light. At that point, I'm going to put it back on safe. I'm going to go ahead and keep them connected, and I'm going to put them back inside my band of leader. At which point, I'm going to grab my spool, and I'm going to be searching for the connector side end, which will be this one with the dust cover on. I'm going to take about a meter, and I'm going to do my first tie down right here on my stake. I'm going to try to put it all the way down to the ground, because there's going to be some tension on there, and I don't want it to be taking the stake with it whenever I put tension. So at that point, I'm going to go back to my, to actually go ahead and put this under a sandbag, the spool. Once the sandbag is secured, I'm going to reach in and grab my tester and my clacker that should be connected, and it should be on safe. And I'm going to also connect my firing wire to it. Now I'm going to perform my second test out here. I'm going to put it on fire. I'm going to couple of creating a shadow right next to my eye and depressing it. I see the light. At that point, I'm going to put it back on safe. I'm going to go ahead and disconnect my wire, replace the cover. I'm going to take off my set, my test set and replace all covers. And that goes as well for my clacker. I'm going to go ahead and put these back in the band of layer. I'm going to secure my firing wire, my spool. And I'm going to ensure that whenever I'm walking down this lane, I have my band of layer always slung. I'm going to go ahead and get up on my feet. And I'm going to begin unspooling this all the way down the lane on my munition side and placement site. As I'm moving, I want to ensure I'm facing the enemy and I am tactically moving with my knees bent. I'm going to get down here. I'm going to set this. I'm going to go on the prone. I'm going to set my spool right here in my workspace where I can see it. I'm going to take off my band of layer. And then I'm going to grab my munition and start in placing. I'm going to go ahead and select about a meter offset from my anchor point right here. I'm going to open up my legs, 45 degree angle, forwards and back before I'm placing it, one meter offset and making sure I am peeking, ensuring that the front towards enemy reading on the munition is actually facing towards the enemy. To that point, I'm going to go ahead and attempt to dig this feet onto the ground. If that didn't work and there was too much gravel, too many stones, then I can go flat 180 degrees, completely flat on the legs and set my mind that way. At that time, I'm going to get about six inches from my munition site and ensure that I can acquire the possible avenue approach where the enemy might be located and aim and make adjustments. At that point, I'm going to go ahead and get my blasting cap out and I'm going to put it underneath a sandbag. I'm going to go ahead and identify the side of my spool, which is running all the way to that anchor point on my firing position. So it'll be this side of the spool. I'm going to unravel about a meter additional to that. And then I'm going to secure my line. I'm going to do a lark's head right here, pull this through, and I'm going to secure that to my stake, ensuring that is snug and all the way to the ground. Now I'll grab my blasting cap, which was underneath a sandbag. I'm going to measure out a meter of slack in this fashion. I'm going to go ahead and hold it with my left hand, with my right hand, I'm going to keep it next to, and right in the center of my chest, with my right hand, I'm going to go ahead and encircle it, allowing it to move up and down, side to side. And I'm going to keep my right hand on my chest, but with my left, I'm going to go ahead and lock it all the way out. And this is my meter mark right here on my right. This blasting cap is going directly back into my spool. Now on this one meter mark, I'm going to make an additional loop from doing the larks head one more time. And I'm going to secure that to my stake, ensuring that it's slowest to the ground as possible and it's snug. Now I'm going to go ahead and arm my munition. So I'm going to grab my blasting cap. I'm going to select the shipping plug that is closest to my stake. I'm going to decouple it. I'm going to go ahead and seat my blasting cap onto my shipping plug on the side that has the slit. Right on top of those threads. I'm going to pull, seat it correctly. I'm going to go ahead and arm it. All right, since I touched my munition, I'm going to go ahead and take aim again, six inches from the site on my munition and adjust accordingly. All right, once this is all connected and my claymore is armed, I'm going to find any foliage and I'm going to camouflage my munition. And also I'm going to camouflage or hide my spool. At that point, I'll ensure that I have all the ventilator contents in here by just peeking out my workspace, ensuring that I have my ventilator slung. I'm going to go ahead and get up. Facing tactically towards enemy, bending my knees. I'm going to technically move back while I am camouflaging my firing line. Once this has come flush, I'll make it all the way back to my firing position. Right here, I'll take the prone and I'll conduct my third and final test. So I'm going to sling this. I'm going to take my tester out. I'm going to go ahead and connect my tester to my clacker. And then I'm going to connect my wire to my tester. I'm going to put it on fire. I'm going to go ahead and create a shadow right next to my eye so I can see the light bulb. I'll depress it. I'm going to say, I see the light. I see the light. I'm going to go ahead and put it on safe. I'm going to go ahead and unplug my cable. And I'm going to replace the cover on the cable. And since I don't need my tester, I'm going to go ahead and disconnect it and put it inside of my bag. So nothing goes on the ground unless it's got a cover. Now I'm going to connect my wire directly to my clacker and I'm going to set off my claymore. I'm going to state claymore, claymore. And then I'm going to put it on fire. And I'm going to depress the clacker. And that will stop my time and that is in placing a claymore. All right, so I'll start my recovery of my claymore mine. It takes about five minutes. It takes five minutes and my time will start right now. So I'm going to go ahead and first thing, I'm going to put it on safe. And then I'm going to go ahead and disconnect my firing cable and replace all covers on both the cable and my clacker. I'm going to go ahead and put my clacker away inside the band layer. I'm going to go ahead and undo my cable. I'm going to then sling my band layer and then start walking tactically to my munition site. And I'm checking for booby traps as I'm traveling down my lane and on mine mine.  I'm checking for booby traps. Now, the first thing I want to do is get my blasting cap secured. So I'm going to do my shipping plug, uncouple my blasting cap and immediately grab my blasting cap and put it inside the slot and my spool. I will then invert my shipping plug to the seal position. Then I'll grab my mine, fold the legs and throw it inside my bag. All right. I'm going to go ahead and undo my first knot, which at this point I want to get rid of this slack. It is a one meter slack for my blasting cap. I'm going to stow that away. Now I'm going to undo my second knot and then I'm going to check quickly on my workspace ensuring nothing is falling out of my bag. And I have all my equipment with me. I'm going to get on my feet and at the same time I'm moving backwards, facing the enemy, I'm spooling back. I want to do most of my spooling out here since once I get to my firing position, I'm going to get on the prone and it's way faster to do it this way. I'm going to get in the prone right here and finish up my spool. I'm going to ensure that both ends are tucked in including the connector site on the slide on the spool.  And once your spool is completely inside, you can go ahead and flap your vandalier, shut and that will stop your time and that's recovery.


I think I will skip almost just because of lack of time. Here is what I'm supposed to say. And as you see, it's very long. And I won't have time, probably, to do everything. So I prefer, sorry? You can take the time. Oh, well, you will see. So the first paragraph is the behavior of vary. If you say, for example, big, very big, very, very big, and you continue, there will be what I call a saturation phenomenon. Adding one more vary won't change the value of what you say. And the language is more clever than all of us together because it can even indicate this saturation point. If you want to say very, very up to the, you say he's really very big, and that means he's very, very and no need to go. You cannot say he's very, really, very big. Really, very big indicates the saturation of vary. And this I can explain to a six years old. He will understand that. But adding one more vary serves to nothing. So I want to see to model this linguistic phenomenon. It's not linguistic because I think it belongs to all languages in French or I want to model this. Of course, the immediate idea is to say, OK, let's call x the set of adjectives to which vary applies. So x is equipped with an endomorphism, which I call T because I'm French and T is the initial of Très. And this saturation phenomenon will be given by, for all x, there exists an n, number n, such that T to the n plus 1 of x equals T of x. Tn, sorry, Tn of x. OK, that's very pleasant but has nothing to do with the language. Why? Because if you write this thing, if there is an n and being well-ordered, there will be a smallest n. Nobody will agree on the smallest n. And I didn't put this in the data. I just said. I mean, the set of fixed points is really good. The fixed points are really good. OK, so certainly this won't work. Insets. So we might try to see, because insets, the natural numbers are well-ordered. OK, so we might try to say instead of sets, let's suppose that x is big x, is an element of a growth nictopause. In that case, n is not well-ordered. And this might work. Well, even in that case, it doesn't work. Why? Because as I said, I can explain this to a five years old boy. He knows nothing about or girl. Sorry, sorry, sorry. I'm speaking English. I had the rather give this talk in French. So you will excuse me if my English is sometimes. So as I said, I can explain this to a five or six years person. And he will understand it or she. But he has no idea at all about natural numbers. He doesn't know what natural numbers are. He doesn't know infinity. This can be said without any use of infinity. Therefore, it cannot be done in growth nictopause. Because in growth nictopause, there is a natural numbers object. OK, so all this is elementary language. So what can we hope for after that? Maybe nothing, maybe this will fail anyway. But we can try to work in an elementary nictopause without natural numbers object. OK? Sorry? The five years old is supposed to know about the nictopause. Well, that was my problem to prepare this talk. Because I need, I shall show how I use them, I need elementary toposes. But nothing has been said here about elementary toposes. And maybe some people don't know what they are or how to use them. So I almost decided to cancel my talk. Because I don't like to speak just for my pleasure and with very few persons understanding. Note that I say persons. OK. So I took this as a challenge. And when I finish, you will tell me if the challenge has been met. So you don't know about elementary toposes. And I don't. I think I defined it in my talk. Sorry? You defined it in my talk and you were not there. Yes, but I mean, you didn't prove anything about what I did. No, it's true. I quoted your Mitchell-Bennabow language and saying that one could use that to. OK, but you didn't. No, OK. But people here, just on the basis of your talk, which I liked very much, could not be able to follow what I say if they don't know how to use the language. So don't panic. The language will be the language of set theory. Sorry? The Mitchell. Yes, OK. I don't like giving names to things. If in the discussion we have the possibility, I'll tell you why Mitchell has nothing to do with that. I have proof of what I say. So it's the naive language of set theory. The difference with the languages that Olivia presented to us is that it is not a first order language. It's a higher order language. You can quantify not only on elements, but on subsets. And for example, the most striking example is one of Peano's axioms, which I will certainly use, namely what characterizes N, many axioms. But the strongest one is that for every subset, so I quantify on subsets of N, if 0 is in S. And if Tx or Tn, when N in S, if S is stable by T, N in S implies Tn in S, T is usually called successor. Then I have all of N. That's the most powerful axiom of Peano's axiom. But you see, it requires the possibility to quantify over subsets, OK? There are, and I shall talk about them, other versions of natural numbers, fried or lovier. And I shall see how they fit with what I do. OK. Now suppose x is equipped with an endomorphism. x now is an object of the elementary topos, which you don't know about. OK. If x is in x, let me first. If S is contained in x, I shall see that S is stable if T of S is contained in S. And now if x is in x, I can take the intersection of all stables of objects, which contain x. And this I shall call the orbit of x. OK. What is this orbit? It's x, Tx, TTx, et cetera. OK. So we have the notion of et cetera. Take the orbit. And now the axiom, which I didn't want to write with sets or with growth in the topos, et cetera, or whatnot, is for all x, the orbit of x has a fixed point. No need to say anything about natural numbers. OK. So the axiom for vary is an endomorphism such that each orbit has a fixed point. If you have objections, you just tell me. I'm confused. Sorry? I am confused. Is this an axiom that you're putting on x? No. I say if I want to express the phenomenon of vary as it is in the language, I can express it by saying for all x, the orbit of x. What do you say for all x? You could have one x, which has a fixed point. Well, I say very has the property that very, very big is very, very, very rich stops. That's the phenomenon. Maybe a source of confusion, Jean. Sorry? That you wrote capital x, the counter power looks like. For all little x? Element of x, right? Small x. Any element of x? Small x. Small x. OK, OK. If I have to write everything, I will never finish. Yes, please. So you're looking at an object having this with equipped with another model, having this property that all objects have a fixed point. That's vary has this. I axiomatize vary by saying it is an object, very, yes, is an endomorphism of an object which has the property that every orbit has a fixed. This can be said in the internal language of a topos. Of course. That's why it needed. This can be said in the internal language of a topos. Since I have a specialist here, he might tell me what if x doesn't have any element. But this is, sorry? I say nothing if x has no elements. And I say the orbit of every element stops. I say nothing. Because x could be empty. And this would be true for the empty. No, even if it's not empty, it can have no global elements. Oh, I'm sorry, x cannot be empty because? No, no, it can be empty. For every, yeah, it can be empty. It can be empty. I don't exclude the empty. Well, the internal language permits, even if x has no global element, to write this formula. But if you don't like these things quantifying on all x where x is something, an element of capital X, which has no elements, let me say it in another matter. Because it will tell you the power of elementary toposes. I shall write for the time being, I shall write, I shall define a relation, which I write like this, x smaller than y, if y is in the orbit of x. And again, what does that mean when there are no elements? But I can define now something which perhaps might satisfy this is a binary relation, as a binary relation. It is characterized by, it is the smallest or the intersection of all pre-orders on x, such that this can be proved. OK, that now this would take intersections, et cetera, et cetera. I could use, and what does it have to do with this? This order relation, because I want to make a remark on the power of toposes. This is a binary relation. A binary relation on x can be viewed as a map from x to the power set of x. OK? And what is this map? It's the map x gives o of x, and now I don't care. This is a binary relation well-defined, so now whether x has elements or doesn't have, I don't care. OK? So I will write as an abbreviation if you want that y in order x, o of x, is an abbreviation of x smaller than y for this relation. OK? Now I think one more remark. Just a side remark. To show the power of elementary toposes, it has to do with families. There are two notions of families, families of sets if you want, or families of anything. A family of sets indexed by i is usually like this, and this notion of family you can do for any category, say with pullbacks because you want substitution. There is another notion of family, which is the one which is used by this set. So I shall call this an implicit family. There is another notion of family. Namely, a family of sets indexed by i is a map from i into sets. Or better, it into some sets of sets. That's the comprehension scheme which says, well, if I have a map from i into sets, I can, well, the power of toposes is that implicit families can always be made explicit. Why? If you have this, intuitively, for each i, you note xi inverse p minus, let's call this p, xi equals p minus 1 of i. And this is a subset of big x. So this tells me this implicit family. I can write it as an explicit family. And this is one of the most fundamental tools of toposes. Implicit families can be always made into explicit ones. OK. So orbits and et cetera, I'll have to be quick on each thing. Orbits and et cetera, orbits are exactly the notion of et cetera, of very simple et cetera. There are more sophisticated et cetera's than this one. But I won't have the time to talk about that. All right. So the idea is to study in a topos an endomorphism by looking at its orbits. I want to understand properties of an endomorphism. Well, first, look at orbit-wise properties, things which are. And then also, how shall I say? See how these orbits fit together. This could be done in sets. It hasn't be done as far as I can see. And many of the things which I will say not be done systematically. Many of the things which I will say I think are new even in the category of sets. OK. So I shall call an orbit an object x together with an element x0 and an endomorphism t, such that t, such that the whole of x is the orbit of x0. That's what I call an orbit. So you want to have that O of x0 is x. That's what you want. That the orbit of x0 is x. The orbit of x0 is x. That's now you call that an orbit. I call this an orbit. That's an abstract orbit. An abstract orbit. Note that x0 need not be unique or anything. No, it's a generator of the orbit. Well, it's a generator. But maybe there are many in how, OK. So. You no longer requiring that every orbit is fine. What properties have such orbits? Are you supposing that the orbit has a fixed point as before? No, no, now I'm talking about orbits in general. I said I want to study an endomorphism by looking an arbitrary endomorphism, by looking at its orbits and then seeing how the orbits fit together. So the first step is what property can an orbit have? First of all, an orbit can be seen as a weak natural numbers object. I shall explain in what sense. Weak. As I said before, there are many notions equivalent of NNO. There are the lovier definition. There are the fried definition, which some people may not know, but I'll recall it. And there is the payable definition. Let's see. Let's compare with lovier's definition. If this is an orbit, and if I have any object y equipped with y0 and translations, I will always call t, OK? If this is an orbit, there is at most 1f making this diagram commutative. So in lovier, there is 1 and 1f. I said weak NNO, replace there exists, 1 and 1s, is there exists at most 1. So this shows how it is a weak NNO in the sense of lovier. In the sense of piano, say. In the sense of piano, it is said that natural numbers satisfy the most important maximum of piano, the fifth of recursion. And you add 0 is not a successor. Let's call s the image of t, s for successor. 0 is not a successor. And the map t is monic. And the big recursion axon. Well, an orbit and also that everything is either 0 or a successor. In an orbit, what do we have from piano's axon? We have the big axon, recursion axon. But we also have that x0 union s by s, I said, is all of x. Another way of saying is that in an orbit, all the elements are either x0 or successors. There are no other elements. So it's, again, a weak. You mean the image of the operator t? Sorry? You mean the image of the operator t? Yes, s I called the image of t, s meaning the successor elements. OK? So again, we see that we have weakened a little bit. Well, a big bit, but nevertheless, we have weakened the definition of natural numbers of piano. Yes? Because this reminds me of the first construction. Can you speak louder? Yes, this reminds me of the construction of Dedekind beginning with an infinite set, and you get the natural. No, no, I mentioned Fried, I mentioned piano, and I mentioned Lovier. I'm not talking about Moore. Otherwise, I will be here tomorrow. What I mean is that if you have 0 is not in s, you have a monolick, so just 0 is not in s. In an orbit, 0 can be in s. In an orbit, 0 can be in s. OK? Well, since you are mentioning this, I anticipate on something which I was going to say anyway, namely this. Suppose I don't even have to talk about orbits. Suppose I have an endomorphism. I shall say that an element x is cyclic, t of x is in the orbit of x. What does that mean? Start with x, write tx, tx, tx, tx, tx, tx. Sometimes you come back to x. I want to say x is in the orbit. x is cyclic, if, sorry, x is in the orbit of tx. That means, I want to understand that, that means start with x, write tx, tx, tx, tx, and you find x. That's correct for cyclic. OK? And let's add one more thing. I shall say that t is acyclic if, not if it doesn't have any cycle. No, its negation doesn't work at all. A is acyclic, if and only if, that's my definition, every cycle, cyclic element is fixed, is a fixed point. I start with x, and then it's just at the very first step I finish. This is acyclic. What about what? That if it has a fixed point. Well, then x is not, if it has a fixed point, every fixed point is cyclic. But acyclic means that the only cyclic elements are the fixed points. That's my definition. OK. Let's call c of x the object of cyclic elements let's call fix x the object of fixed elements. This one is very easy to describe. No need of higher order, it's just fix x is the equalizer of the identity and t. But cyclic elements, you cannot define in a category unless it is a toposso or toposlike. We have always this inclusion. And to say that t is acyclic is this equality. OK. Let me just, I know everything, but I might forget a few elementary things. Now, first theorem is that an orbit is acyclic if and only if the pre-order relation on the orbit is an order relation. OK. Have to be proved, it's not difficult to come to difficult question a little bit later. For example, the natural numbers, if they exist, are acyclic because the order relation induced by successor is an order relation. Well, again, we shall see, I said, orbits are weak natural numbers. Let me give another example. If is an orbit, then it is a monoid with one generator. And such monoids are commutative. So when you prove that the natural numbers is a commutative monoid for addition, et cetera, and has a generator, that's true for any orbit. So many things, when proves for natural numbers, are true for orbits or other, for example, for acyclic orbits, then you have a total order. OK. I rush. The fourth thing is killing the cycles. It's passing to the associated order. How do you do that? If you have a pre-order relation, you can take the associated order. Let's write x equivalent to y if smaller than y and y smaller than x. And let's write x doubly equivalent to y if x and y belong to the same cycle. These things are almost the same. When is x equivalent to y if and only if x equal y or x and y are cocyclic? This relation is symmetric, transitive, not necessarily reflexive because x and x need not be cocyclic and yet they are equal. So we can take the quotient of x by this equivalence relation. This quotient is now t respects this equivalence relation. So factors as an endomorphism of this. However, this is an order relation. The pre-order becomes an order relation on x. And the orbits now, each orbit of x gives an orbit here. But now the orbits are ordered. Therefore, this is a cyclic. So what we have done is just by identifying two points which are in the same cycle, we have replaced the whole cycle by a fixed point. And this says what? Let's call by, to have notation, t of e, the category of objects equipped with an endomorphism. Let's call a of e, the subcategory of this, formed only with a cyclic thing. What I have just proved is contract an adjoint by just killing the cycle. Killing means reducing them to a single element. And in particular, if every orbit of t of x has a cycle, then for the associated cyclic object, every orbit will have a fixed point. And we will be in the situation of vary. There are many more things to say, but I have to take only a tiny bit of each paragraph if I want to respect the time. So what comes next? There is a difficult theorem which says the following. This one is really the other ones are little exercises each one takes maybe 10 lines at most, provided you give them in the correct order. But here is a difficult theorem. An orbit is finite if and only if it has a cyclic element. That's a fact. Why? When you say finite, I'm sorry, I don't want to annoy you, but just for the purpose of the talk, can you tell how you define finite? Finite is Kuratovsky finite. There is a notion of finite. But again, if you don't know elementary toposes, think of finite as finite. I want to speak things that people can understand. Sorry? If you don't have the action of choice, I'm not saying anything about the action of choice. In an elementary topos, one defines Kuratovsky finite elements, which in the case of sets are just the finite elements. In the case of Grotendijk toposes, they are a bit more complicated. But think of them as finite. That's all I can do. It would be good for the purpose of your talk. That if someone write down the notions of Kuratovsky finite on the blackboard, maybe you don't want to do it, but Olivia is ready to do it. I don't care. Provided she doesn't take on my time of talk. Well, it is, in a sense, first it is difficult, but it has some philosophical connotations. That means that this abstract definition of finiteness, or even in set, the definition of finiteness, has something to do with orbits, with, et cetera. Finite has something to do with, et cetera. Orbits are axiomatization of, et cetera. So I think I don't take your time of talk. You are. So the idea is this, is that one can describe using how you call the logic, the collections of finite subset of a power set. Because what you have are the singletons. The singletons are the singleton mass. This is given. And finite subset are generated by finite union of singletons. So you take the smallest joint sub lattice. I mean, sub object of p is closed under union, and which contain the singleton. And if you do that, you get all the Kuratowski finite subset of its. Kuratowski finite subset of its is something which is contained in the smallest, and x would be Kuratowski finite. If x itself is contained in the smallest. So the joint is also there. OK. Well, let me say I don't like your definition. I shall give an equivalent definition. No, no, why? It's not your definition. By the way, it's not your definition. It's the definition we use. OK, so I shall give another definition which will perhaps make it more intuitive why there is something between orbits and finiteness. Instead of saying that this is a joint sub lattice, that means we can suffice to say that it's stable by the following operation. If s is in power set of x and x in x, then it's stable by not taking the union of two things, taking the union of one thing and a singleton. You obtain all of x by iterating the process of taking the union of x and the singleton. What does it have to do with, et cetera? Well, I didn't want to talk about this, but the et cetera I mentioned is a very simple one. There are more complicated, et cetera. For example, suppose we have x equipped with a family of endomorphisms, not just one. The et cetera will consist in taking something, the orbits are obtained by taking something, and ti, any i, of that something, not just this is a kind of multiple, et cetera. Well, the power set of x has this structure. Namely, if x equals the power set of i, it has this structure by taking ti of a subset with i in i equal the subset union i. We don't want to interrupt a lot, I'm sorry. No, no, you don't have to push it. You don't have to push it. I added some time. So what's a finite object? Or more generally, we look at the orbits in the sense which I will not define of this ti. The orbits are all the finite sub-objects of i, saying that i is finite is that, OK? I mean, we have one definition so that you can explain your theorem now. No, the theorem is this. An orbit is finite if and only if it has a cyclic element. And that's a difficult theorem. How can you see this? Because I want you to see to have a vague idea of what's going on, but it's not trivial. Suppose you have a computer and a repetitive process. Start with one thing, take ti of that thing, and then ti, ti, et cetera, et cetera. And suppose you give the computer science, by the way. So don't. And suppose you give him the only instruction, start with some x, repeat the computation, and stop when you find something you have already, a result you have already met. And it stops, which means ti, et cetera, et cetera, has a cycle. It will stop, certainly, in a finite number, in a finite time. Conversely, if it stopped in a finite time, that means that it has met the cyclic element. That is very vague. But that's how it is. Except that in that case, it's finite in the usual sense of set theory, whereas k finite has been defined by you, now, or by other persons. But the point is that this finiteness, for example, doesn't have some properties one might expect of finite. For example, a sub-object of a finite object need not be finite. We all believe that the subset of the finite has set this finite. Of course, because we live in set theory in a Boolean topos. I'm not supposing it's Boolean. Here is another very strange and, to me, it was surprising result. Here it is. Suppose x is smaller than y. We can, that is y in the orbit of it, we can define the closed interval x, y as being all the z, which are between x and y. Now, the following are equivalent for all these intervals our finite is equivalent to all the xx, our finite. This can be proved. But this equivalence is not true in general. It is true if and only if the topos is Boolean. Now, here we get if and only if the whole structure of the topos is determined by this. And of course, if t is acyclic, these two equivalent conditions are satisfied. But if they are not acyclic, the topos has to be Boolean. That says also something. Acyclic means maybe there is no fixed point. If there is a fixed point, OK, it's acyclic. But maybe there isn't. For example, for the natural numbers, there is no fixed point. However, every interval of the natural numbers, closed interval, is finite. We know that for the natural numbers. But we know it now for every acyclic orbit. Not only the thing which goes all the way, the orbit which goes all the way to the fixed point. If there is a fixed point, it's finite. But all the intervals are finite. But the surprising thing is that Booleanness is needed unless you assume the orbits. The thing is acyclic. Booleanness is needed in general for this equivalence. So again, some properties of et cetera imply properties of the whole topos. Now, seven is connected components. In this case, I shall give an idea of the proof because I like it and because it doesn't take a long time. We have this order relation. So it's a category. So we can talk about the connected components. When are two objects, two elements, x and y, I shall put another equivalence thing in the same connected components? The answer is very simple. Namely, they are in the same connected component if the orbit of x and the orbit of y has a common element.  It's not so obvious unless I miss something. Well, this relation is obviously reflexive and symmetric. It remains to show that it is transitive. Usually to get the connected component, you have to make zigzags like this. But here that says just one zigzag is enough. Why? But suppose x, y, z. There is an element u in the orbit of x and there. And there is an element v here in the orbit of z intersect the orbit. But these, u and v, are both in the orbit of y. And we have seen that in an orbit, the pre-order is a total pre-order. So we have either u in the orbit of v or v in the orbit of u. But then that's trivial to check that it is transitive. OK. So pi 0 is here is one more thing, one more application of the orbits. Suppose we have x equipped with an endomorphism. I want to talk about t, t composed with t, t composed with et cetera. Well, you look at x to be x, and it is equipped with an endomorphism. Let's call t blank for each f take t composed with f. So I can take the orbit of the identity. And the orbit of the identity is exactly what we want. So we have not only a level of elements, but of functions. For example, suppose you want to, suppose you say, OK, with nature numbers, I can write for all x, there exists an n such that tn plus 1 of x equals tn of x. But suppose I want to say something more. Namely, not for all x, there exists an n. But there exists an n for all x, which is good enough for all x. So you see I'm writing formulas as if there were nature numbers, but there isn't. How do you say this? That means the orbit of t has a fixed point. And this will give you a uniformity of t in x to the power x. For all x, such that for all x, this holds. I want to be able to say that this fixed point, there is a uniformity, a uniform bound. So you see that this notion of et cetera covers many, many, many things, even when there is no nature numbers object. And you want to do as if there was one. So this orbit, the orbit of the identity along t, I shall call it the leading orbit. And what property does it have? For every x, the leading orbit maps subjectively on the orbit of x. Take the evaluation. If you have, say, f, well, if you have in the orbit of t in the leading orbit, let me call it all x, the leading orbit, then for each x, evaluate f at x. And this is a mapping of this into of little x. And we know that such mappings are unique. And we know they are subjective. So every orbit of any element is a quotient of this leading orbit. For example, if this leading orbit has a fixed point, then all orbits have a fixed point. Now I will finish with just one more remark, important one. I have said that very has the property that each orbit has a fixed point. OK? I claim that this is a very important notion, mathematical notion. What does it mean? That t is such for all x has a fixed point.  Why? Each connected component is a rooted tree, which means all the orbits are finite, have a fixed point, et cetera, et cetera. That's the correct definition of a tree. So it's important. This says that very is a rooted forest. I think I have. Any questions? So during the lecture, it was mentioned about. If you will, please speak louder. I'm hard of hearing. OK. So you mentioned the definitions of natural number objects in a topos. You mentioned the names of Raid, Lovir, and Piano. Is it possible to give the three definitions precisely? So I want to know the three definitions of. An elementary topos? Well, I suppose it is an elementary topos. I'm not sure. The actions of Piano in a topos. The actions of Piano in a topos are the actions of Piano that everyone knows. It's rather the actions of the natural numbers. There's Lovir, it's a mini d'anandomorphism S, and an element 0, which is universal for this property. And for the orbits, what I showed you is that there is no existence, but there is unity. If it exists, there is unity. Instead of having existence, there is unity. The actions of Piano are 0 is not a successor. The successor is a mono, and everything is 0 or a successor. Which is very nice. In one moment, when I look at the successor and the element 0, it's a sum. It tells me that 0 is not a successor because it's joint. And it tells me that everything is a successor. And it also tells me that a successor is a mono, because in a sum, it's a mono. It says all this. And what's left of it? It's the fact that, although it's not said like that, that Grantin is a connex. What is said by saying that... We don't say it, it's what we write. The co-equalizer, it means exactly that Grantin is a connex. All orbit is a connex. Very good. Because each element can be connected with the starting point of orbit. What's left of it? That's Freud. I said to hear it. Piano, I've already mentioned it many times. It's not worth losing time. There's a paragraph that I haven't mentioned, but which will be very good in the discussion. A plea for the language. Pleadoyer for the language. And it will go very well. That's already a plea for the language.


This 2019 Jeep Grand Cherokee Limited in Sangria, Pearl Coat is stock number 21J150A. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used Jeep and Jeep Grand Cherokee headquarters. This vehicle has a 3.6 liter Pentastar V6 motor. It has been fully safety and inspected by our service shop. Has a fresh oil and filter change. All the fluids have been checked and topped off. And this Jeep is 100% ready to go. Once again Sangria, Pearl Coat is the color. If you want to subscribe to our YouTube channel to see all the daily videos that we do in the lower right hand part of the screen is a gold subscribe button. Click that to see all the daily videos of new and used vehicles. And if you have HD capabilities on your computer, tablet, smartphone or television turn them on right now because it is definitely your best way to check out the quality and condition of this particular Grand Cherokee. Before seeing it in person. Comes with the 18 inch painted and polished Luma rims. And it has Goodyear Assurance max life tires on here. These tires have a really good amount of tread left. I would say probably 80, 90%, maybe even a little bit more. Very new tires. These are 265, 60 R18s. Front fender is in excellent shape. I didn't see any dents or dings on there. Comes with the projector lamp headlamps. Front bumper is in nice shape as well. I didn't see any major scuffs or scrapes or cracks. And the hood is in excellent condition as well. Passengers side front fender. No dents or dings on there. And the passenger side rim is in nice shape as well. No scuffs or scrapes on there. As you go down this side of the vehicle, take note of how clean the body is, how reflective and mirror like that paint is. We take these HD videos so if you are far away or even if you're close by and you just cannot make the trip down but you're still interested in purchasing the vehicle, you can see the vehicle, hear the vehicle, and of confidence in what you're looking at before you even get here. So when you do get here, all you have to do is drive it, love it, and maybe all set to go. Very clean down that side. Didn't see any dents or dings. And this back rim is in excellent shape as well. The back tires have just as much tread as the front tires. And coming around to the back of the vehicle, rear bumper looks great. It does have the backup parking sensors, full towing package which includes receiver hitch, four pin and seven pin wiring. And the tailgate is in pretty nice condition. It does have a power lift tailgate and the LED running lights on the back here. Back storage area is nice and clean. No rips or tears and it comes with a full size spare tire and jack doors. Those seats do fold down for extra storage and this back hatch, press this button. It'll beep at you three times and then begin its descent. Actually beeps at you a total of four times, but on the third one it comes on down. And as you go down this side of the vehicle, just as clean as that passenger side, no major dents or dings on there. And for full disclosure, this back rim is in nice shape as well. That tire is very new as well. Down the rest of this side, no dents, no dings. It does have the blind spot monitoring, built in directional signals, heated mirrors, enter and go system. Just have to have the key fob in your pocket and it'll automatically unlock inside the limited package. Gives you the black leather interior. There are no rips or tears on the seat. Really nice shape. Power driver seat with lumbar. Both these front seats I believe are heated. Comes with factory formats, auto headlamps. You get the nice dark wood grain trim on the dash and that wraps all the way around the door with the chrome accent. Power windows, power locks, power mirrors and you do get a memory driver seat. We'll hop in here and turn the ignition on so you can see the miles and check out the radio and everything. You can see that this one has 41,466 miles. You do get a compass display up there, digital speedometer. And you can change that to that or that. Really a nice setup with that 7-inch TFT display. You get the heated leather wrap steering wheel. Cruise controls on the right. Bluetooth and information center controls on the left. And it does have audio controls on the back. This one comes with the 8.4C radio which gives you AM FM and Sirius XM radio capabilities. You also get the factory navigation system on here. You can do your climate controls here as well. You notice you do get dual climate controls. You can also do your heated seats and heated steering wheel. And you can check out that backup camera at any time. Now those grid lines, they are dynamic which means that they turn as you turn that steering wheel. And that center line, of course, lines right up with your receiver hitch to get you hooked up to your boat trailer or camper the first time every time. You also have all your different apps on here which includes the projection manager where you can project your cell phone to the screen via Android Auto or Apple CarPlay. We actually did a demo on Android Auto on the upper right-hand part of the screen. Down here is your climate controls. Your more tactile ones. Your backup parking sensors. Start-stop capabilities. Eco mode, sport mode, and your stability control. 8-speed automatic transmission with the mechanical shifter. And down here you get your Quadra Track 2 system which gives you snow, sand, auto mud, and rock. There is also a neutral setting for flat towing. Downhill assist control and four-wheel drive. Low, passenger side format and seat are in nice shape. Rips or tears on there. And the headliner is in nice shape too. I didn't see any rips or tears on there. Home link buttons for garage door security systems and lane systems. SOS and assist buttons in the rear of your mirror. And we'll check out the back seats. And then we will start it up and check out under the hood. Back seats are just as clean as the front seats. No rips or tears back here. Does have the latch child safety system for any child car seats you may want to try and put back here. Factory floor mats back here as well. You do get a 115 volt, 150 watt plug-in, two USBs.  Also comes with side curtain airbags. You get that nice wood grain trim and chrome accents back here as well and the child safety locks on the doors. We'll start it up and take a look under the hood. Starts right up. No check engine lights or anything like that. I would personally like to thank you for checking out the video today and hopefully from this HD video you will be able to tell just how clean this Jeep is all the way around inside and out. Under the hood we have the 3.6 liter Pentastar V6 motor. Engine bay is super clean. It's very smooth. Once again the Jeep has been fully safe and inspected by our service shop because they fresh oil and filter change all the flutes have been checked and topped off. It's in the beautiful Sangria pro cap. Kind of got a little bit of purple-ish to it. I'd highly recommend this vehicle to make quality and condition standpoint and to see more pictures of this Jeep for one of our other 450 new and used cars, trucks, SUVs, minivans, Wranglers, Grand Cherokees, Cherokees, you name it. We got to go to that website right there www.SummitAuto.com Full pictures and descriptions of every single vehicle www.SummitAuto.com If you'd like to check out more HD videos you can go to www.youtube.com Remember to like, subscribe and share on this video and all the videos that you see there. In fact, in a second you'll see a link to subscribe to our YouTube channel in the upper left. A link to more Grand Cherokee videos like this one in the upper right. A link to this vehicle on our website in the lower left. And a link to one of our latest YouTube videos in the lower right. Click those, check us out. And we're super excited to help you with this Ultra Clean 2019 Jeep Grand Cherokee Limited in Sangria pro coat.