My Math Forum Limits' and reals' cardinality

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 February 20th, 2018, 11:41 AM #11 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 "You are talking the same old unmitigated nonsense again."* is not a refutation of Cantor's Diagonal Argument. * v8archie, last post Last edited by zylo; February 20th, 2018 at 11:46 AM.
 February 20th, 2018, 12:02 PM #12 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,509 Thanks: 2514 Math Focus: Mainly analysis and algebra No, but then I'm not trying to refute it.
February 20th, 2018, 12:57 PM   #13
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 Originally Posted by v8archie No, but then I'm not trying to refute it.
Would someone else be justified in using your argument to refute it?

What amazes me is that a nonsensical statement passes unnoticed, and buries a serious contribution, but when I attempt to respond to a serious question to my thread in "Real Analysis," Real Numbers and Limits, the thread is closed.

February 20th, 2018, 02:15 PM   #14
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 Originally Posted by Maschke Hi Loren. I'm not sure exactly what you're getting at. Can you provide some context? Limits in the real numbers are defined by quantifying over uncountable sets. When we say, "for every epsilon ..." and then give the rest of the modern formal definition, we are implicitly quantifying over uncountably many potential epsilons. This has philosophical and historical implications when we are talking about how the limit concept replaced the old idea of infinitesimals. __________ You use the phrase, "this difference, the limit function minus f(a) itself ..." and asked if this is a cardinality. But it's a NUMBER, not a set. [Numbers are ultimately sets, but not in this context!] For example let $f(x) = 0$ if $x \neq 0$, and $f(0) = 47$. In other words when you input anything other than $0$ into the function, you get back $0$. But when you put in $0$, you get $47$. Now in this case the limit of $f$ as $x \to 0$ is $0$. But $f(0) = 47$. So the difference of the limit of the function at that point, and the value of the function at that point, is $47$. It's a number. Does that make sense? Remember, a function doesn't even need to have a value at a given point in order to have a limit there. __________ Well, the cardinality of the continuum is the problem that's been driving set theory since Cantor first asked the question. Our usual axioms don't tell us the cardinality of the real number continuum, and the hunt for new axioms hasn't resolved the question. Some think that asking about the cardinality of the continuum is not a meaningful question. Other set theorists have esoteric approaches such as Ultimate-L and the set-theoretic multiverse. [Those are ideas I've read a little about. I would not want to give the impression I know what they mean]. These are very deep waters. Nobody has any idea what is the cardinality of the continuum. We know it's $2^{\aleph_0}$, but nobody has any idea what cardinal that is. __________ Does this relate to your previous question about space-filling curves? They are generally surjections but not bijections. They hit some points more than once. There's a bijection between the real numbers and the unit square, if that's what you mean by a finite surface. You can get a bijection of the unit interval to the unit square by interleaving the digits of the decimal expansions of the coordinates of the points in the square. Then you can biject the unit interval to the reals. And you have to wave your hands at the .4999... = .5 problem. There are only countably many of those pesky things so you can either ignore them or sit down and figure out how to deal with them explicitly. Is anything I wrote helpful? I'm not sure what is the intent of your questions.

Maschke, thank you for taking the time to answer my questions directly and thoroughly. You would do well teaching undergraduate math majors. You essentially answered my questions, written or surmised, and brought forth new lines of thought. I can largely understand your interpretations. You got it.

Keep on topic. You all see most of my intent although without formal descriptions from me, and contribute what is condensed by Maschke in most helpful form.

Btw, what significance does "47" have?

 February 20th, 2018, 02:25 PM #15 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory Is "interleaving" anything like the diagonal argument?
February 20th, 2018, 02:31 PM   #16
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 Originally Posted by Loren Maschke, thank you for taking the time to answer my questions directly and thoroughly. You would do well teaching undergraduate math majors. You essentially answered my questions, written or surmised, and brought forth new lines of thought. I can largely understand your interpretations. You got it. Keep on topic. You all see most of my intent although without formal descriptions from me, and contribute what is condensed by Maschke in most helpful form. Btw, what significance does "47" have?
So glad I was able to hit the mark on this one. 47 is my favorite random-sounding integer.

Quote:
 Originally Posted by Loren Is "interleaving" anything like the diagonal argument?
Interleaving works like this. Say you have a point $(x,y)$ in the unit square, and say that the decimal expression of $x$ is $.x_1 x_2 x_3 \dots$ and the decimal expression of $y$ is $.y_1 y_2 y_3 \dots$.

Then you can map the pair $(x,y)$ to the real number $.x_1 y_1 x_2 y_2 x_3 y_3 \dots$. Likewise given the decimal expression for some real number in the unit interval, you can reverse the process to get a pair of reals representing the coordinates of some point in the unit square.

You can do the trick for any $n$-dimensional Euclidean space. Cantor was shocked to discover this. He thought each dimension up gave you a higher cardinality. Turns out to be false. The line, the plane, 3-space, 4-space, etc. all have exactly the same cardinality.

Last edited by Maschke; February 20th, 2018 at 02:47 PM.

 February 20th, 2018, 03:11 PM #17 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory Does the uncountability over the set of epsilons correspond to a particular cardinality? Can the set of cardinal numbers (ordinal numbers?) itself belong to a set with its own cardinality (ordinality?), or is this "circular" reasoning? Do we know somethings about "2 to the Alef" by being able to define a "greater" cardinal set? Where can I find examples of cardinal number sets (like that of the set of all curves in space)?
February 20th, 2018, 06:11 PM   #18
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 Originally Posted by zylo Cardinality of the reals: countable.
So we're back to this are we? There's no point in discussing it, as you have consistently shown.

Last edited by greg1313; February 21st, 2018 at 12:34 PM.

February 20th, 2018, 09:42 PM   #19
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Quote:
 Originally Posted by Loren Does the uncountability over the set of epsilons correspond to a particular cardinality?
Yes, the amount of epsilons is $2^{\aleph_0}$. Again, this is our only way to express it. We don't know what cardinality this is, except from this.

Quote:
 Can the set of cardinal numbers (ordinal numbers?) itself belong to a set with its own cardinality (ordinality?), or is this "circular" reasoning?
The cardinal numbers/ordinal numbers do not form a set. The circular reasoning you mentioned is valid and very important. It is a paradox called the Burali-Forti paradox. The only way out is not to let the cardinal/ordinal numbers be a set. So they're only a what is called a proper class.

February 21st, 2018, 08:52 AM   #20
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 Originally Posted by greg1313 So we're back to this are we? There's no point in discussing it, as you have consistently shown. If you continue to post off-topic nonsense you will be banned.
The OP subject is cardinaity of the reals and limits, which I directly addressed.

Why can't I express my opinion like everyone else in this thread? What are you afraid of? Part of what I have said I said about 1000 posts ago. Has nothing in this thread ever been said before?

Quote:
 Originally Posted by zylo The real numbers are given by the unique limiting sequence (not the limit of the sequence), as n approaches "infinity," of n place decimals, which can be counted for all n. Cardinality of the reals: countable. EDIT: Limits of the reals are a subset of the reals. .3333...3m, any fixed m, to n places of 3's, has the limit as n approaches infinity, 1/3
In
Real Numbers and Limits
I wrote
"REAL NUMBERs are defined uniquely by "infinite" (unending) sequences of natural numbers. The sequence IS the real number.
LIMIT is a defined property of real numbers."

which is a simple, clear, logical, transparent foundation of real analysis, which is why I posted it in the Real Analysis forum.
This thread was closed when I tried to post an answer to a question.

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