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Axiom of Infinity and Reduction

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aleph_1

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GrandMinnow,

I truly appreciate your comments. When I read what you have to say, I may be assured that it is interesting and true. Thank you.

Let me also say that

1) You would be remiss in offering only your correct response, "No", to the boy without

2) Giving Paul Cohen mention, but

3) You have already conceeded that our understanding of infinities involves whim, and so

4) I conclude that there is a rotten apple in the bowl.

By the way, I did discuss Cantor's Theorem and its proof with him, as well as the theorem that the countable union of countable sets is countable, and the proof that there are only a countable number of stars in the universe. You can construct the latter proof yourself.

Getting back to the Axiom of Infinity, you have given me much to mull iver already. That is one reason I appreciate you.

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(1) I don't claim that the mere answer 'no' provides an explanation. But an explanation does start with pointing out, as I did, that the aleph operation is on ordinals.

 

(2) It's not needed to mention Paul Cohen in order explain what the alephs are and why "aleph_(1/2)" makes no sense.

 

(3) I don't know what particular remarks of mine you have in mind. I've touched a bit on how I view set theory, and I did reply to one of your comments about 'whim', but nothing I've said should be construed as saying one way or another whether "our understanding involves whim". I don't even put the matter in such terms. 

 

(4) What other apples DO you like? I.e., is there a formal theory you do approve of that axiomatizes the mathematics for the sciences? 

Edited by GrandMinnow
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Guest Math Bot

This is over-complicating the issue to insane degrees.

Infinity is not a really complicated concept.   Infinity simply prefers to the potentiality to continue in a sequence to which "infinity applies".  It means that although you have stopped *here*, the sequence could be theoretically be extended, and that no matter where you do stop, that is always going to be the case.   You have to stop in the sequence at some point, but no matter where you do stop, the sequence does not end there.  It means that however far you get in the sequence, it is part of the sequence, and that the sequence has further units contained within it.  But in actuality you can progress in the sequence only so far, even though you will not exhaust the potential number of units within the sequence. 

For instance, you can say that there are an infinite number of natural numbers, an infinite sequence of natural numbers, if one chooses to consider the set as a sequence of mathematical units.  No matter how long you count them, you will not run out of them.  If you could keep counting, you could potentially find a representation of the next natural number.  The natural numbers one, all the way up to 1x10^6777, are contained within the set of natural numbers, but so are the natural numbers 1x10^6777 up to 1x10^999999, and indeed any range of natural numbers ever conceived, the sequence of listing them could be continued without limit, *except that one has to stop doing so eventually*.

Edited by Math Bot
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Your answer demonstrates that you do not understand the Axiom of Infinity (AoI). Suffice it to say, AoI does not relate to the limit concepts you have presented. It amounts to the existence of an integrated unit that is actually infinite. Once you cede ground to the actually infinite, it is a small step to producing infinities of differing sizes via Cantor's Theorem. This is the realm of rationalism unreduced to reality and pertains to the analytic-synthetic dichotomy. In this case, the rationalism seems harmless because it leads to standard mathematics which is manifestly useful. However, even the likes of Brouwer and Kronecker were uncomfortable with the state of the theory, and Godel and others commented on it but were apologists of sorts for the current theory in some ways.

There are other problems, like impredicative definitions, but I thought that AoI would be a good place to start. Ye know them by their fruits. Either we are content to have unreducible notions in our theories or we are not. Unreducible notions are dragons. You can define them (flying repriles that breathe fire) but you must ultimately reject the concept since it is unreducible to anything observable. Is the existence of different sizes of infinity reducible? If not, then there are dragons afoot.

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Oism permits integration of an arbitrary number of units into concepts, and then those may be integrated into units of other concepts. As already pointed out by GM, the issue that I have may result not from the axiom of infinity, but from the power set operation, or even the notion of "set" itself.

Reduction only comes into play when (reasonable?) Processes of integration and differentiation result in different infinities, the distinction of which cannot be reduced to perception. Reduction of the set of natural numbers is not necessarily a requirement of Oism, as far as I can tell. The process of forming the set of natural numbers is one of integration of numbers as units of the concept "natural number". That there are an infinite number of units of this concept may or may not be a problem for Oism.

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From ITOE2 pg. 16-17

"A concept is like an arithmetical sequence of specifically defined units, going off in both directions, open at both ends and including all units of that particular kind. For instance, the concept "man" includes all men who live at present, who have ever lived or will ever live. An arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means only that whatever number of units does exist, it is to be included in the same sequence."

 

The same would apply to the natural numbers. Their arithmetical sequence extends into infinity, without implying that infinity actually exists; such extension means on that whatever quantity of nautral numbers does exist, they are to be included in the same sequence.

 

To quip Corvini:

"It gives us the sense that we can extend the (number) sequence without limit (i.e. infinitely), because we know exactly what to do (just add another symbol (number) to the left, representing 10 of the group unitized to its right), in order to accomplish it."

 

Eventually, you would run out of ink, or symbols, or the abiltiy to recall what the symbols stand for, or paper to write it upon, or computing power, etc. Thus, as MathBot aptly pointed out, once the process is stopped, the actual result is always finite.

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Guest Math Bot

No it does not demonstrate that at all, it demonstrates that the concept of infinity is much more simple than you and mathematicians might like to pretend.  I understand what they say, I would see no reason to over-complicate the issue and conceptualize infinity in such a way.

One can disagree with the formulation of something, without failing to understand that formulation.

Edited by Math Bot
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No it does not demonstrate that at all, it demonstrates that the concept of infinity is much more simple than you and mathematicians might like to pretend. I understand what they say, I would see no reason to over-complicate the issue and conceptualize infinity in such a way.

One can disagree with the formulation of something, without failing to understand that formulation.

Except, the OP is not about this conception of infinity at all. A set is finite if there exists a bijection between that set and an initial segment of the natural numbers. A set is infinite if it is not finite. This is simple enough and accepted without question. The question here is about the existence of a single entity that is infinite. Do infinite entities exist? Surely, they are not reducible to anything since reduction is finite.

These considerations lead to questions about not only the nature of reduction but of integration. Numbers are already abstractions and integration is unlimited. I.e., one may integrate an unlimited number of units into a concept. Is there a difference between the concept of natural number and the "set" of natural numbers? The Power Set Axiom does not apply to the concept "natural number" but applies to the set. One can accept the concept of natural number without accepting the axiom of infinity (the existence of the "set" of natural numbers. Are there different versions of reduction and integration one can identify here?

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Can we separate that which "leads to standard" mathematics and is hence useful, and that which does not? 

 

Also, just because you "can" do something with some sort of mathematics does NOT mean you have to.  That is a choice each Mathematician makes.  So for example if cardinal numbers are useful (I may have that wrong) in some respect but because of their definition you open up infinities of different sizes which may not in fact be useful, then perhaps it is up to the Mathematician to know what to do with the cardinal numbers (or whathaveyous) and what not to do with them... and perhaps choose a different specialty.

 

 

Then again it is hard to know what will and will not eventually be useful in mathematics.  Wasn't the solution of Fermat's theorem very involved (non elegant) with widely varying types of mathematics (elliptical functions or something or other)? 

 

I think some sort of standard regarding the "probability" that some thread will be made back to more basic and useful problems could help ground a mathematician.  That said I think it is very difficult to say any particular branch of pure mathematics is utterly useless and/or disconnected from reality or anything in it.  

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Sorry for not staying on topic.

 

 

Infinite sets always seemed to me to extend a specific easily visualizeable concept, "a" collection of discrete things, the collection being something capable of being "a" "thing" and extend it into "some-like-a-thing" quite beyond concrete visualization. 

 

I think there is a BIG step between conceiving of  let x be any integer and thinking of "a thing" which is ALL the integers.  It's the difference between: 

 

ANY number

 

and

 

ALL numbers.

 

try applying THAT kind of thing in the process of measurement omission.

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Guest Math Bot

Except, the OP is not about this conception of infinity at all. A set is finite if there exists a bijection between that set and an initial segment of the natural numbers. A set is infinite if it is not finite. This is simple enough and accepted without question. The question here is about the existence of a single entity that is infinite. Do infinite entities exist? Surely, they are not reducible to anything since reduction is finite.

These considerations lead to questions about not only the nature of reduction but of integration. Numbers are already abstractions and integration is unlimited. I.e., one may integrate an unlimited number of units into a concept. Is there a difference between the concept of natural number and the "set" of natural numbers? The Power Set Axiom does not apply to the concept "natural number" but applies to the set. One can accept the concept of natural number without accepting the axiom of infinity (the existence of the "set" of natural numbers. Are there different versions of reduction and integration one can identify here?

 

If you understood what I was saying about infinity, you would grasp infinities cannot exist, except as a potentiality for the extension of a sequence beyond any given point in which you stop progressing within in. 

Clearly entities cannot in any sense be infinite.  The concept applies only to the abstract potentiality to continue within a sequence.  NOTHING ELSE.

Edited by Math Bot
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Clearly entities cannot in any sense be infinite.  The concept applies only to the abstract potentiality to continue within a sequence.  NOTHING ELSE.

Aleph is talking about sets, not series or sequences. An infinite set is different than an infinite series. So, Aleph is asking about how one could reduce an infinite set to reality in order to be a valid concept. You did it in terms of a series, great. Can infinity also be validly applied to sets? Your position seems to be no, meaning you agree with Aleph. I suppose one thing to ask is about the difference between a set and a series. A set is discrete, a series is continuous. When referring to a set (I think) you'd be referring to all the members, while with a series you're not necessarily referring to numbers per se only their progression. I'm still learning about all this, but thinking about these threads interests me.

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Guest Math Bot

Aleph is talking about sets, not series or sequences. An infinite set is different than an infinite series. So, Aleph is asking about how one could reduce an infinite set to reality in order to be a valid concept. You did it in terms of a series, great. Can infinity also be validly applied to sets? Your position seems to be no, meaning you agree with Aleph. I suppose one thing to ask is about the difference between a set and a series. A set is discrete, a series is continuous. When referring to a set (I think) you'd be referring to all the members, while with a series you're not necessarily referring to numbers per se only their progression. I'm still learning about all this, but thinking about these threads interests me.

I know that he is talking about infinite sets, but I am telling him what infinity is, a concept which applies equally to infinite sets.  I am also telling him entities are not infinite.  Maybe you are both too dense to notice. 

The only valid sense in which infinity applies is that you can keep on enumerating elements in some sets.  Entities are finite, progressions can be infinitely extended.  There is a big difference between the two.  Sets are finite in actual size, but infinite in potentially.  This is an important distinction. 

Edited by Math Bot
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I know that he is talking about infinite sets, but I am telling him what infinity is, a concept which applies equally to infinite sets.  I am also telling him entities are not infinite.  Maybe you are both too dense to notice. 

The only valid sense in which infinity applies is that you can keep on enumerating elements in some sets.  Entities are finite, progressions can be infinitely extended.  There is a big difference between the two.  Sets are finite in actual size, but infinite in potentially.  This is an important distinction. 

Therefore you would argue that there is no such thing as a valid infinite set. Which is fine; that's what Aleph is asking about. I've had ideas though of how an infinite set can be valid, though, but I'm not clear enough on several points. I need to research more before I can state my case. Still, the axiom of infinity applies to sets, and intuitively, it would seem that *of course* sets can't be infinite because they have cardinality. And as far as I've seen, that is why Aleph thinks that infinite sets are metaphorical dragons.

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Take two sets of objects, say a pile of blue pebbles and a pile of red pebbles. How does one establish that there are as many blue pebbles as red pebbles if this fact cannot be ascertained by merely observing the two piles?

 

If this is approached by the method of pairing, one takes a unit from each pile and sets them aside and repeat the process. The question is not resolved by the new paired piles, the undiscovered answer remains in the original piles yet to be subjected to the pairing process. Only when the process is complete or reaches the point where the remainder of red and blue pebbles are perceptually the same can the answer be determined.

 

Can this be accomplished with an "infinite" set? Paraphrasing Corvini, the answer could be yes, . . . if the problem is rigged.

 

Since the process of pairing is not resolved until one gets to the end, and the idea of infinite as MathBot, Aristotle and Corvini suggest is the continued extention of the process, if we ask if we have an infinite number of individuals in a room, do we have the same number of heads, or hearts, or noses - assuming that all the individuals are normal.

 

To apply this to the integers and the even integers - we could establish the sets as follow:

(1, 2, 3, 4, . . . n)

(2, 4, 6, 8, . . .2n)

 

Each of these sets can be extended as required. Infinite in this sense is that the relationships remain true to whatever extent you extend it. You can conceptualize it as such because you know whatever integer you can come up with, multiplying it by 2 will give you the counterpart even integer. The fact that we can do this for any integer extends to the fact that we can do this for all integers, and gives the impression that there may indeed be an actual infinity - but is there? What is the infininth integer? What do you get if you add one more to it. Infinity just means that you can continue to apply the process of adding one more to whatever you have, and do it again. As Aristotle points out, when you stop adding one more, the result is still finite.

Edited by dream_weaver
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Guest Math Bot

Therefore you would argue that there is no such thing as a valid infinite set. Which is fine; that's what Aleph is asking about. I've had ideas though of how an infinite set can be valid, though, but I'm not clear enough on several points. I need to research more before I can state my case. Still, the axiom of infinity applies to sets, and intuitively, it would seem that *of course* sets can't be infinite because they have cardinality. And as far as I've seen, that is why Aleph thinks that infinite sets are metaphorical dragons.

 

There are sets that are finite in size, but infinite in extension.  This is the key difference.  All sets are finite in size at any given moment, they have so many identified elements, but in some cases there is the potential to identify more, that is they are infinite in extension.  But they must be finite in size.

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