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Why the universe cannot logically be infinite in time backwards

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File:Wooden hourglass 3.jpg
passage of time, imaged/S. Sepp

Further to “No Big Bang: Universe Always Was” just posted by Donald McLaughlin, Ashby Camp gave Uncommon Descent permission to post these notes from a class he taught at the 2018 Harding University Bible Lectureship titled “Answering the New Atheism.” Worth pondering:


1.The second premise of the Kalam cosmological argument is: The universe began to exist. It is more reasonable to believe this is true than to deny it because, Scripture aside, there are strong philosophical and scientific reasons for believing it.

a. The philosophical argument for the universe having a beginning is that past time cannot be infinite because an infinite amount of time cannot already have been exhausted so as to arrive at the present. Infinite time is limitless, inexhaustible, and thus cannot have been exhausted.

(1) Put differently, one could never traverse an infinite sequence of time units, an infinite number of seconds, minutes, hours, etc., to arrive at now. There always would be more time units to traverse before now. If one begins counting down from minus infinity, one cannot count to the present. An infinite amount of time can never pass because it is limitless; it can only be in process, never complete. To quote the New Dictionary of Christian Apologetics, (p. 700), “One can neither count from one to infinity nor count down from infinity to one. There is always an infinite distance to travel, so one never arrives.”

(2) Sean McDowell and Jonathan Morrow make the point this way in Is God Just a Human Invention? (Grand Rapids: Kregel Publications, 2010), 75-76:

Imagine you went for a walk in the park and stumbled across someone proclaiming aloud, “… five, four, three, two, one—there, I finally finished! I just counted down from infinity!” What would be your initial thought? Would you wonder how long the person had been counting? Probably not. More likely, you would be in utter disbelief. Why? Because you know that such a task cannot be done. Just as it’s impossible to count up to infinity from the present moment, it’s equally impossible to count down from … infinity to the present moment. Counting to infinity is impossible because there is always (at least) one more number to count. In fact, every time you count a number, you still have infinite more to go, and thus get no closer to your goal. Similarly, counting down from infinity to the present moment is equally impossible. Such a task can’t even get started! Any point you pick in the past to begin, no matter how remote, would always require (at least) one more number to count before you could start there. Any beginning point would require an infinite number of previous points. Here’s the bottom line: we could never get to the present moment if we had to cross an actual infinite number of moments in the past. Yet, since the present moment is real, it must have been preceded by a finite past that includes a beginning or first event. Therefore, the universe had a beginning.

(3) The impossibility of infinite past time, of having already traversed an infinite timespan, does not mean that future time will not go on forever. Future time is potentially not actually infinite. In other words, it is infinity in progress, something that will move toward infinity but never arrive; you’ll never get to the end of it. So it poses no problem like the claim of having already traversed an infinite timespan.

(4) The impossibility of traversing an infinite timespan, an infinite sequence of time units, need not mean that God has not always existed. There are ways of dealing with God’s relationship to time so that he does not exist (or has not always existed) in a sequence of individuated moments, a sequence of time units. His eternality is not one of infinite time but one of either timelessness or a different kind of time that has no measure or metric. For example, William Lane Craig’s view is that “God is timeless without creation and temporal since creation.” Philosophers Alan Padgett and Richard Swinburne refer to time before creation as “metrically amorphous time,” meaning it differs from our “measured time” (see, e.g., Eternity in Christian Thought).

(5) This philosophical claim that the universe cannot always have existed makes sense to me and to many philosophers, but there are others who are not persuaded.

(a) Some, for example, point to the fact a finite timespan, say one minute, can be subdivided infinitely into units of decreasing length, and yet one can still traverse that timespan. The claim is that in going from 0 to 1 minute one traverses an infinite number of time units to arrive at 1 minute, so it is not true that one cannot traverse an infinite number of time units to arrive at the present. But there is a problem with that claim. The subdivisions of a finite timespan are only potentially infinite in number. It is true that one could keep subdividing forever, but each further subdivision results in a finite number of subdivisions the sum of which is the finite timespan being subdivided. The number of subdivisions can grow toward an infinite number but can never actually reach an infinite number. Whereas, when speaking of an infinite timespan one is speaking of an actually infinite set of time units, the sum of which is an infinite length of time.

(b) Though this philosophical claim that the universe necessarily began to exist continues to be debated, it dovetails nicely with the scientific acceptance of the universe having a beginning. That is the subject to which I now turn.

  

Readers?

Comments
If Granville agreed with Jerad then what is posted in comment 12 doesn't say that. But then again not one of my detractors knows how to read for comprehension.ET
March 15, 2020
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JVL:
That’s exactly right, you are just matching up the elements regardless of their ‘value’.
Learn how to read. You are NOT matching the elements.
You disputed it but you cannot point out a fault with it.
LIAR. All I have done is pointe out its faults.
AND it’s widely used.
It is NOT use for anything. It is a useless concept.
Of course you don’t like it!
It's not that I don't like it. It's BOGUS, you ignorant loser. Both set subtraction and my counter example contradict Cantor. That you refuse to understand is a reflection on YOUR willful ignorance. And that all you can do is repeat what I am disputing proves that you are a willfully ignorant coward. Good luck with that. Let me know whenever you are able to think for yourself and can formulate an argument without using the disputed nonsense.ET
March 15, 2020
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Jimmie:
Ed, you have to admit, ET’s just as good at biology as he is at math.
And I am better than you at both.ET
March 15, 2020
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Jim Thibodeau: Dang JVL, you’ve made someone very angry. I'll struggle on, somehow. None of the ID scholars here disagree with ET on math, so I have to assume it’s of the same piece. Who knows? I'm sure Drs Dembski and Sewell agree with me. Do you suppose ET is a Darwinist who’s just trying to make ID people look like angry fools? No, he's sincere.JVL
March 15, 2020
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ET: I SAID IT IS BOGUS, IE THE WRONG TOOL FOR THE JOB. I WOULD EXPECT ALL COUNTABLY INFINITE SETS TO BE ABLE TO BE PUT IN A ONE TO ONE CORRESPONDENCE. THAT IS BECAUSE THAT IS JUST A POSITION THING. That's exactly right, you are just matching up the elements regardless of their 'value'. Well done! You got it. You can stop shouting now. Right. You told me to use the methodology that I am disputing. The methodology that set subtraction proves is bogus. You disputed it but you cannot point out a fault with it. AND it's widely used. You lost that dispute. Your scheme is bogus. Of course you don't like it! It shows that you are wrong. But that doesn't make it incorrect. And I’m right. You cannot find a one-to-one correspondence using the matching scheme used to determine unions, subsets and proper subsets. Which is why I didn't use it. I used a method that DOES exhibit the matching I'm trying to prove exists!! It's allowed. Check out any website or textbook you like; you can use any scheme you like because it's all about the number of elements NOT their values. So again, all Jerad can do is use the methodology I am disputing, to try to settle the dispute. You have serious issues, Jerad. Just because Jerad and his minions are unable to grasp what that means just proves how clueless they are. You lost the dispute because you can not find fault with it except to shout that it's bogus. There is no dispute just you not wanting to admit you made a mistake. The world of textbooks are rife with examples of things being taught one way only to have that over turned. Jerad thinks that knowledge has to be stuck in the 19th century. And whines because I do not. Not in mathematics. what was true 2000 years ago is still true, what was true 1000 years ago is still true, what was true in the 19th century is still true. You really do not understand how mathematics works. I have acknowledge I was wrong many times. It isn’t my fault that you can only make your case by using the very thing I am saying is bogus. Because it's not bogus, you lost the 'dispute'. Because what the gave us was and is used. And proved true with real world applications. No one uses the concept that all countably infinite sets have the same cardinality. The concept doesn’t even make any sense. Can’t compare Euclid and Pythagoras to Cantor’s useless concept. YOU don't use it so you think it's useless. That doesn't make it so. AND, regardless if there are read-world applications, that does not affect its truth.JVL
March 15, 2020
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Do you suppose ET is a Darwinist who’s just trying to make ID people look like angry fools?Jim Thibodeau
March 14, 2020
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None of the ID scholars here disagree with ET on math, so I have to assume it’s of the same piece.Jim Thibodeau
March 14, 2020
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Ed, you have to admit, ET’s just as good at biology as he is at math.Jim Thibodeau
March 14, 2020
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Eddie "I know absolutely nothing" George. :) :) :)ET
March 14, 2020
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ET
I have acknowledge I was wrong many times.
:) :) :)Ed George
March 14, 2020
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Cheerleader Jimmie, waving his pom-poms. And lying. Always lying.ET
March 14, 2020
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JVL @ 125:
What was true for Pythagoras and Euclid is still true.
Because what the gave us was and is used. And proved true with real world applications. No one uses the concept that all countably infinite sets have the same cardinality. The concept doesn't even make any sense. Can't compare Euclid and Pythagoras to Cantor's useless concept.ET
March 14, 2020
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Dang JVL, you’ve made someone very angry. http://intelligentreasoning.blogspot.com/Jim Thibodeau
March 14, 2020
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I have acknowledge I was wrong many times. It isn't my fault that you can only make your case by using the very thing I am saying is bogus.ET
March 14, 2020
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You’re not even trying to answer my matching question.
I SAID IT IS BOGUS, IE THE WRONG TOOL FOR THE JOB. I WOULD EXPECT ALL COUNTABLY INFINITE SETS TO BE ABLE TO BE PUT IN A ONE TO ONE CORRESPONDENCE. THAT IS BECAUSE THAT IS JUST A POSITION THING.
One baby panel showing one simple situation and you think it’s all okay?
I considered who I was providing it for.
I told you how to deal with your “dispute” and you failed.
Right. You told me to use the methodology that I am disputing. The methodology that set subtraction proves is bogus.
You not being able to find an unmatched element between the positive integers and the positive even integers under my scheme shows you can’t back up your claims.
Your scheme is bogus. And I’m right. You cannot find a one-to-one correspondence using the matching scheme used to determine unions, subsets and proper subsets. Why do we have to use yours when set subtraction contradicts it? So again, all Jerad can do is use the methodology I am disputing, to try to settle the dispute. You have serious issues, Jerad. Just because Jerad and his minions are unable to grasp what that means just proves how clueless they are. The world of textbooks are rife with examples of things being taught one way only to have that over turned. Jerad thinks that knowledge has to be stuck in the 19th century. And whines because I do not.ET
March 14, 2020
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ET: Yes, set subtraction proves that your scheme is bogus. You aren’t right. You are not even wrong. You're not even trying to answer my matching question. That says it all really. I linked to the website to prove that set subtraction is both useful and used. You lose. One baby panel showing one simple situation and you think it's all okay? hahahahahahahahahah Clearly you do. All you can do is keep referring to the very thing I am disputing as if that settles the dispute. And I am sure that anyone reading this understands that what you are doing is foolish. Anyone who can use logic and reasoning, anyway. Because you haven't responded to the situation! I told you how to deal with your "dispute" and you failed. So the dispute is over. Set subtraction refutes your claim, Jerad. And just repeating your pap isn’t going to change that fact. You not being able to find an unmatched element between the positive integers and the positive even integers under my scheme shows you can't back up your claims. And you've had plenty of time. You keep running in hopes the truth won't catch up with you. And I’m right. You cannot find a one-to-one correspondence using the matching scheme used to determine unions, subsets and proper subsets. Here we go AGAIN! You think some matchings are allowed and some aren't but you've NEVER been able to back that up. You keep recycling your unsupported arguments. So here we are- we will just be repeating that for a while. ???? I'm good with you not being able to find an unmatched element between the positive integers and the positive even integers under my scheme, which is entirely fine. I'm good with being correct. I'm good with you ignoring the math, ignoring the arguments, ignoring the answers to your objections. But you've run out of room to run. Time to admit you've made a mistake. (I know you won't admit you're wrong and you'll keep this going INTO infinity because you just cannot acknowledge that you are ever wrong.)JVL
March 14, 2020
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And I'm right. You cannot find a one-to-one correspondence using the matching scheme used to determine unions, subsets and proper subsets. So here we are- we will just be repeating that for a while. :razz:ET
March 14, 2020
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Yes, set subtraction proves that your scheme is bogus. You aren't right. You are not even wrong. I linked to the website to prove that set subtraction is both useful and used. You lose.
I don’t need any help;
Clearly you do. All you can do is keep referring to the very thing I am disputing as if that settles the dispute. And I am sure that anyone reading this understands that what you are doing is foolish. Anyone who can use logic and reasoning, anyway. Set subtraction refutes your claim, Jerad. And just repeating your pap isn't going to change that fact.ET
March 14, 2020
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ET: Using a bogus scheme.And I showed you how the odds are matched! A bogus scheme? Are we back to your unsupported "contrived" stuff again? We are talking about the same matching used in a Wikipedia article. And you showed me how the odds are matched? Maybe you should just go lie down and give it a break. Being generous and assuming you meant you showed me that the odds are unmatched I have CLEARLY shown, twice, how they are matched. Try again. You are a liar. I showed and told you how to do it. No, I'd remember you coming up with a "relative cardinality" of the primes. You didn't do it. Anyone can scan through the thread and see that. You're starting to sound like Donald Trump, i.e. you're making things up which the rest of the world knows is wrong. It’s used. There isn’t anything preventing anyone from using it on sets with infinite elements. You can use it. It doesn't work with cardinalities but you can use it. Funny that no one else does. Ooooo, you linked to an elementary school level Venn diagram! Is that really the best you can do? That didn't address cardinalities at all. You really are grasping. Why would I use the WRONG TOO, FOR THE JOB? Clearly you are an ass Because you cannot admit you are wrong. And you cannot identify unmatched elements between the positive integers and the positive even integers under my scheme. And don't keep claiming you provided a list because I dealt with that and you were wrong as can be clearly seen. No one cares. Why hasn’t anyone come to your defense? Your two cheerleaders are just ignorant trolls. I don't need any help; what I am saying matches what every textbook, every paper on set theory and every online resource says. And you can't find any unmatched elements from the positive integers and the positive even integers using my matching scheme. You are a liar and a toll, Jerad. But I'm right. And you cannot find any unmatched elements between the positive integers and the positive even integers using my scheme.JVL
March 14, 2020
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JVL:
And I showed you how the odds are matched!
Using a bogus scheme.And I showed you how the odds are matched!
No, you did not.
You are a liar. I showed and told you how to do it.
Repeating the same thing over and over again is not support.
And yet that is all you do.
Well, you’ve been unable to provide any support for “set subtraction” so . .
It's used. There isn't anything preventing anyone from using it on sets with infinite elements. set subtraction
Find some unmatched elements in my matching.
Why would I use the WRONG TOO, FOR THE JOB? Clearly you are an ass
Why then has no one come to your defence on this thread?
No one cares. Why hasn't anyone come to your defense? Your two cheerleaders are just ignorant trolls. You are a liar and a toll, Jerad.ET
March 14, 2020
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ET: I provided a list!!!! Clearly you are a liar. And I showed you how the odds are matched! Are you even paying attention? Yes, you bare sorry. And yes I told you how to do it. No, you did not. Which has a higher cardinality: the positive multiples of 7 or the positive prime numbers? Of course I have Repeating the same thing over and over again is not support. You don’t know that. Well, you've been unable to provide any support for "set subtraction" so . . . And I am not the one who uses the concept being disputed to try to settle the dispute. You are that fool. I told you how to win your "dispute" and you can't manage to do that. So there is no dispute. It still remains that absolutely no one uses the concept that all countably infinite sets have the same cardinality. It’s as useless as the people saying it. That’s you, Jerad. Not true but it doesn't affect its truth. Not according to set subtraction. Find some unmatched elements in my matching. It's not the odds, I showed you how those are matched. Of course they do. They see an impotent person, who is unable to deal with the reality that ID has the science and he can’t refute it. So that person has to pick a fight over a topic he doesn’t understand. Really? Why then has no one come to your defence on this thread? Why is it that kairosfocus, bornagain77, martin_r, News, Barry Arrington, Granville Sewell and all the rest have left you out to sway in the wind? Even they don't agree with you. And what I am disputing is not basic set theory. They see that you can’t even get subtraction right! What set theory course did you take? What was the course number? Which university was it? Who was the author of the textbook?JVL
March 14, 2020
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JVL:
But you can’t say what they are!!
I provided a list!!!! Clearly you are a liar.
Sorry but you did not say whether the multiples of 7 or the primes had a larger cardinality.
Yes, you bare sorry. And yes I told you how to do it.
You have provided no support for what you are saying.
Of course I have
If it works it’s funny no one uses it for infinite sets.
You don't know that. And I am not the one who uses the concept being disputed to try to settle the dispute. You are that fool.
I’ve told ;you over and over and over again how to settle your ‘dispute’ but you keep failing to do so.
Yes, you keep referring to the very thing being disputed. It's as if you are proud to be a dolt. It still remains that absolutely no one uses the concept that all countably infinite sets have the same cardinality. It’s as useless as the people saying it. That’s you, Jerad.
It’s true regardless.
Not according to set subtraction.
Of course the rest of the world doesn’t see it that way.
Of course they do. They see an impotent person, who is unable to deal with the reality that ID has the science and he can't refute it. So that person has to pick a fight over a topic he doesn't understand. And what I am disputing is not basic set theory. They see that you can't even get subtraction right!ET
March 14, 2020
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ET: Set subtraction uncovers unmatched elements. Period. End of story. But you can't say what they are!! Too funny. I already covered the odds so you've got to find another answer. And I have already explained how it works with the sets Jerad mentions. But that is moot as Jerad can’t even grasp the basics. Until that happens there is no way he will be able to grasp anything else. Sorry but you did not say whether the multiples of 7 or the primes had a larger cardinality. YOUR MATCHING SCHEME IS BOGUS. What part of that are you too stupid to understand? Set subtraction uncovers the unmatched elements. What part of that are you too stupid to understand? You have provided no support for what you are saying. That means it's only you saying so. Find an unmatched element or admit you can't. The odds are matched. I showed you how. It works! Only a desperate loser says that it doesn’t. Math books tell you that dividing by 0 doesn’t work. So clearly Jerad is making things up because he can’t handle the truth. If it works it's funny no one uses it for infinite sets. Puzzling that. Do you really want to talk about dividing by zero? I'm happy to do so. And I am not the one who uses the concept being disputed to try to settle the dispute. You are that fool. I've told ;you over and over and over again how to settle your 'dispute' but you keep failing to do so. Which means you can't. It still remains that absolutely no one uses the concept that all countably infinite sets have the same cardinality. It’s as useless as the people saying it. That’s you, Jerad. It's true regardless. And if Jerad being a cowardly ass makes UD look foolish, then so be it. I say it reflects more on Jerad than it does on UD. Of course the rest of the world doesn't see it that way. The rest of the world says: look at that UD site, they can't even get basic set theory right.JVL
March 14, 2020
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And if Jerad being a cowardly ass makes UD look foolish, then so be it. I say it reflects more on Jerad than it does on UD.ET
March 14, 2020
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Set subtraction uncovers unmatched elements. Period. End of story. And I have already explained how it works with the sets Jerad mentions. But that is moot as Jerad can't even grasp the basics. Until that happens there is no way he will be able to grasp anything else. YOUR MATCHING SCHEME IS BOGUS. What part of that are you too stupid to understand? Set subtraction uncovers the unmatched elements. What part of that are you too stupid to understand?
Of course not, no one is going to waste time discussing something that doesn’t work!
It works! Only a desperate loser says that it doesn't. Math books tell you that dividing by 0 doesn't work. So clearly Jerad is making things up because he can't handle the truth. And I am not the one who uses the concept being disputed to try to settle the dispute. You are that fool. It still remains that absolutely no one uses the concept that all countably infinite sets have the same cardinality. It’s as useless as the people saying it. That's you, Jerad.ET
March 14, 2020
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ET: And set subtraction proves your scheme is bogus! Which means there would be some unmatched elements but you can't find any. That’s Cantor’s point with respect to different infinities. I don't think he mentioned different densities. Try again. Set subtraction is a real thing and it really works. Show how it works with these two sets: Let S = the set of the positive multiples of 7 and P = the set of positive prime numbers. I suspect you will ignore this but people will notice. t works just fine. And I have explained it to you. You are just an obtuse loser. You haven't showed it working with those two sets. And AGAIN- set subtraction proves your scheme is bogus. So just repeating yourself proves that you are an ass. Ignoring my answers doesn’t mean they weren’t provided. You keep running away from answering a very long standing question: ;you have yet to find an unmatched element in my matching scheme. So the sets must the same size. Which means 'set subtraction' fails. End of. So if you are going to ignore what set subtraction uncovers and just keep repeating your nonsense, then there isn’t anything else to say. IF subtraction is correct then you should be able to find some unmatched elements. But you can't so subtraction fails. End of. Each one has infinite elements. However, each has different densities of elements. Please give a rigorous mathematical explanation of what this means.
The cardinality of the reals is greater than the cardinality of the integers, i.e. the real numbers are NOT countably infinite.
Yes, I know. That is what differing densities means.
So . . . the cardinality of the reals is greater than the cardinality of the integers? By how much? Again, Jerad isn’t anyone to say that set subtraction cannot be applied to sets with infinite elements. There isn’t anything in any math text book that supports that lie. Of course not, no one is going to waste time discussing something that doesn't work! AND you haven't found any reference supporting it. And you keep not answering the same question . . . Cantor’s proof is for checking if the two sets are countably infinite. But being countably infinite doesn’t say anything about the cardinality. You know what? Since no one is taking you seriously I'm not sure why I should keep arguing with you . . . Except that I think it makes a site like this look completely idiotic to not call one of its most prolific contributors on promulgating ridiculous notions. You looking foolish doesn't matter that much. Uncommon Descent looking foolish hurts the cause. A lot. I bet people on other sites are laughing at you and UD right now.JVL
March 14, 2020
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Again, Jerad isn't anyone to say that set subtraction cannot be applied to sets with infinite elements. There isn't anything in any math text book that supports that lie. Cantor's proof is for checking if the two sets are countably infinite. But being countably infinite doesn't say anything about the cardinality.ET
March 14, 2020
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Each one has infinite elements. However, each has different densities of elements.
Nope, wrong answer.
No, it's the right answer.
The cardinality of the reals is greater than the cardinality of the integers, i.e. the real numbers are NOT countably infinite.
Yes, I know. That is what differing densities means.ET
March 14, 2020
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JVL:
But I can find matchings with no elements left out!
And set subtraction proves your scheme is bogus!
Please link to some source which explains what that means.
That's Cantor's point with respect to different infinities.
It doesn’t work with some sets though does it?
Set subtraction is a real thing and it really works.
Like comparing the set of all multiples of 7 and the prime numbers.
It works just fine. And I have explained it to you. You are just an obtuse loser. And AGAIN- set subtraction proves your scheme is bogus. So just repeating yourself proves that you are an ass. Ignoring my answers doesn't mean they weren't provided. So if you are going to ignore what set subtraction uncovers and just keep repeating your nonsense, then there isn't anything else to say.ET
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ET: Set subtraction uncovers unmatched elements. But I can find matchings with no elements left out! I covered the odds so . . . Set subtraction proves my point. You haven't found any unmatched elements in my scheme. There is only ONE infinity. It can be populated with differing densities of elements. Please link to some source which explains what that means. Each one has infinite elements. However, each has different densities of elements. Nope, wrong answer. The cardinality of the reals is greater than the cardinality of the integers, i.e. the real numbers are NOT countably infinite. But you studied set theory (what was the textbook used by the way, you haven't answered yet) so you knew that. But again, that is well over your limited thinking ability. I'm happy for you to explain what that means mathematically. Some references would be nice. Subtraction is the basic arithmetic operation used to determine if one thing is larger than the other. It doesn't work with some sets though does it? Like comparing the set of all multiples of 7 and the prime numbers. No help there. It doesn’t get any more foundational than that. So yes, I am going to question anyone who goes against what subtraction uncovers. And the “answer” I am getting is subtraction doesn’t work with sets of infinite elements- cuz it don’t, so there. Seriously. Whatever. I did give some examples where it doesn't work but I guess you ignored those. I'm just thinking you haven't been able to find any unmatched elements in my scheme for the positive integers and the positive even integers. If you can't find one then the sets are the same size. And because I have called out that explanation as total nonsense, I am the one getting something handed to him. Dodging the fact that there is a long standing question you haven't been able to answer. How does that work? 'Cause you keep dodging questions and ignoring answers? People are really starting to notice and I don't really intend to embarrass you any further but . . . Here's a matching scheme for two infinite sets: Let J = the positive integers, let E = the positive even integers. Match them up with the following scheme: "1" from J is matched with "2" from E "2" from J is matched with "4" from E "3" from J is matched with "6" from E "4" from J is matched with "8" from E "5" from J is matched with 11.23 from E. Just kidding, "5" from J is matched with "10" from E "6" from J is matched with "12" from E "7" from J is matched with "14" from E "8" from J is matched with "16" from E "9" from J is matched with "18" from E "10" from J is matched with "20" from E "11" from J is matched with "22" from E and so on. Now, I may be wrong but it seems obvious to me that if you continue on in the same fashion that ever element in J is matched with one and only one element in E. And every element in E is matched with one and only one element in J. No element of either set is left out. In fact, if you specify a specific element of either set I will be able to tell you its "partner" in the other set. Both sets are well defined (I can easily tell you if something is in the set or not). The matching I am proposing matches the criteria listed well above for this sort of thing. And I keep asking . . . Given that matching the only way J or E is larger than the other is if there are some unmatched elements. So anyone who wants to say the two sets are not the same size only needs to find some elements that are unmatched. That's all they'd have to do. So, can anyone find an element of J or E that is unmatched? Yes or no?JVL
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