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Irrational numbers and Physical constants

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Not a -stolen- concept but an idealization.  There can only be a countable number of real numbers that can be computed to any desired degree of accuracy.   However,  that leaves the real number field full of holes unless we assume some property that guarantees local compactness.   Most mathematicians prefer a number field that  has no holes  rather than one in which a Cauchy Sequence does not converge to a member of the field.   Being able to take limits makes calculation a lot easier.  

 

That is the good news.

 

The bad news might be that physical reality is not locally compact and may be full of holes at the lowest end of resolution.

 

ruveyn1

Replace compactness with completeness and your comments make sense. I understand your confusion because in the context of functional analysis we often seek to impose a topology under which some set becomes compact so that limits don't lie outside of the space in question. Here we are talking about the completion of the rationals to form the reals.

Limits are a big question--right on point ruveyn1. Can you part with limits? Is the notion of limit the issue or the construction of the number system? I claim that the classical construction of the number system was done so that we could have an agreeable (and simple) notion of limit. The two issues are linked.

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"The Continuum Hypothesis may be phrased that every uncountable set . . ."

Uncountable? Please elaborate on what that means in this context. If uncountable = infinite here, then again the question is "can there be different size infinities." Real things are finite though. There really can't be different size infinities merely because there can be no infinities at all in reality. Logic is a matter of how real things work. If the idea of non-finite things, things which don't exist, leads to only illogical conclusions, that doesn't matter as far as reality is concerned.

 

After that, I'm really not even clear what the question is to begin to try to answer it. It would be really nice to not have mathematical jargon used where it isn't necessary. "Equinumerous" for example (using that for the example since that one is not hard to figure out what it means based on the parts of the word) could instead have been said something like "has the same number of things contained in it as". Examples would be great too.

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"The Continuum Hypothesis may be phrased that every uncountable set . . ."

Uncountable? Please elaborate on what that means in this context. If uncountable = infinite here, then again the question is "can there be different size infinities." Real things are finite though. There really can't be different size infinities merely because there can be no infinities at all in reality. Logic is a matter of how real things work. If the idea of non-finite things, things which don't exist, leads to only illogical conclusions, that doesn't matter as far as reality is concerned.

 

After that, I'm really not even clear what the question is to begin to try to answer it. It would be really nice to not have mathematical jargon used where it isn't necessary. "Equinumerous" for example (using that for the example since that one is not hard to figure out what it means based on the parts of the word) could instead have been said something like "has the same number of things contained in it as". Examples would be great too.

Cantor proved there is no 1-1 onto map from the set of integers to the set of real numbers.  But the integers are a proper subset of the set of real numbers.  Ergo there are fewer integer than real numbers.  A humble example.  A is a pile of ten object B is a pile of 20 objects we can map A into a proper sub set of B  but there is not 1-1 map between A and B.  Since A matches a subset of B  but B cannot match A  we conclude that A has fewer objects than B.   This kind of (mis) match can be extended to transfinite sets.

 

ruveyn1

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Is that so?

 Fantastic.

 What is this, an attempt at a "Gotcha"? A mental entity is not an entity independent of consciousness. For a mathematician you (surprisingly) seem to enjoy contradicting yourself.

(1) Not an attempt at a "gotcha". So that the poster may respond to clarify, I merely highlighted that the poster's own argument seems to go in the opposite direction he seems to intend. (2) I've already said that, as far as I understand, Objectivism holds that a mental entity is not independent of consciousness. (3) I'm not a mathematician. (4) I don't know what contradiction you believe you have found in my posts. 

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Minnow, in the appendix to ITOE Ms. Rand clarified the use of "mental entity". The most accurate term is mental existent.

 

I read where she uses the term 'mental entity' and says that concepts are mental entities. I don't wish to insist that Objectivist terminology does or does not prefer 'mental existent', but reading Ayn Rand in a straightforward manner would seem to allow using the term 'mental entity', just as she mentioned that concepts are mental entities. 

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you seemed to have some objections with the terms Grames used

I didn't object to his terminology. I just wished to say that, as far as I understand Objectivism, it doesn't reject the notion of sets. While, of course, as I've said, I do understand the point that Objectivism would not accept that such abstractions as sets have existence independent of consciousness. And, yes, I quite understand your own point in that regard too.

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I apologize to bluecherry for my lingo. I just want to know the place of reduction when it comes to math. Is it necessary to reduce all concepts to perception or can one maintain a system philosophically separate from perception that is purely rationalistic but that has correspondences to perception. For instance, we know that continuous population models are invalid but we still use them because they are useful.

I cannot answer many of your objections now, but let me say this concerning the Continuum Hypothesis. In mathematics, we say that two sets are equinumerous if there is a one-to-one and onto (bijective) correspondence between them. Yes, they have the same number of elements: cardinality. An initial segment of the natural numbers is a set of the form {1,2,3,...n} for some n. A set is finite if it is bijective with an initial segment of the natural numbers. A set is Infinite if it is not finite. A set is countably infinite if it is bijective with the entire set of natural numbers. The natural numbers are countably infinite since it is bijective to itself. A set is uncountably infinite if it is infinite but not countably infinite.

The Continuum Hypothesis basically says that there are no infinities between the cardinality of the entire real numbers (the continuum) and the countable infinity. That is because any uncountable subset of the real numbers must have a cardinality that is less than or equal to that of the real numbers, being a subset. The CH says that it can't be "less than".

All of these concepts are well-defined, but many are not reducible to perception. This is an opportunity to see whether reduction is possible or desirable is the process of constructing a rational notion of number.

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Unless you can find an Objectivist treatment of mathematics that goes as far as such matters as complete ordered fields and such, I don't see the point of even asking Objectivists such questions.

To fail to address this issue is to concede that Objectivism is not productive as a philosophy. It is as dead as its founder.

Edited by aleph_1
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To fail to address this issue is to concede that Objectivism is not productive as a philosophy. It is as dead as its founder.

Is that so? Well then, what does the competition have to say about this? What is the Post-Modernist, Marxist, Existentialist or Analytic-Linguistic position on cardinal infinities?

Rhetorical question. They have nothing to say about it either because it is not a philosophical question.

As far as the question of finding what the idea of infinity reduces to, I think that has already been answered. Infinity reduces to the open-ended nature of concepts.

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Uncountable?

Going off ideas I had based on of what GrandMinnow explained (and if you can clarify if what I say is consistent, GrandMinnow, I'd appreciate that), it seems to me like a countable sets include all the types of numbers that can be counted in some consistent, known way. So, natural numbers go on forever, but certainly countable as n+1. If I give you a natural number, you'll always know what's next. 1000, 1001, etc. I don't know if this applies to rational numbers, or other numbers. What's next after 3.543? Is it 3.544? Or 4.543? Any notion of next doesn't make sense, so extend it into infinity, and you might not know what the next number is, and even measuring cardinality of a set with those numbers is a problem. You'd have to define the pattern better. With just decimals, it's not so bad, but get to complex numbers, prime numbers, irrational numbers, then the problem gets worse.

 

Real things are finite though. There really can't be different size infinities merely because there can be no infinities at all in reality.

I think you're saying that since there are no actual infinities, only potential infinities, it wouldn't make sense for infinity to have varying sizes. Justice doesn't exist walking around "out there" and it wouldn't make sense to ask if one justice is bigger than another justice. A numerical comparison between abstractions that have no physically measurable parts is hardly sensible. Certainly there is a metaphorical size, but no actual size. I can say if a tree has a bigger height than another tree, because they are actual; concretes are being compared.

But I'm doubtful that such an approach is right. Concepts I think can be rightfully considered infinite sets, because an infinite number of members (referrents) can be put into a concept, and forever into past and future. For some concepts, a size between abstractions makes sense. I'd say mathematical concepts are the main type of concept where a size comparison can be made. A number basically abstracts everything about an existent except mere existence, so a "3" is just 3 objects of any size, shape, color, or anything else for that matter. Comparing small numbers could be perceptual, and a simple version of comparing tree height.

With big numbers, how could you say 5000 is smaller than 90000? There must be abstract comparisons, and you are unlikely to find an actual 90000 of anything.

Still, that's somewhat concrete. At levels of abstraction of algebra and beyond, which are derived from those simpler numbers and mathematical concepts, there won't be any concretes. Is 5000x bigger or smaller than 90000x? It's smaller of course, but you don't even need any concretes to answer the question. The value of x isn't even needed. Basically, any expressions (5000x, 90000x) are patterns that are bigger or smaller, at least when abstracting from numbers/math. Infinity is something I barely understand, I just see it as a way to capture how abstractions and patterns go on forever without regard to time. At the very least, two infinities can be of different sizes even if uncountable, provided that their patterns as sets differ in magnitude, cardinality, or some comparable way.

If there are illogical conclusions, that would only mean the pattern in question is an absurd possibility, a floating abstraction. Of course, the thing to do is reduce a concept down to see where the problem occurred. I suppose the illogical conclusions don't matter in a literal way, as long as the person tries to resolve the contradiction.

Edited by Eiuol
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I think we may conclude that Oists philosophy is inconsistent with the Infinity Axiom since that axiom implies the actual existence of an infinity. From this we may construct Natural Numbers, but not "the set of Natural Numbers". We may also construct Rational Numbers, but not the "set of Rational Numbers". There is, consequently, no way to define the countable infinity, since there can be no bijection with a set that does not exist.

How one constructs Real Numbers is an outstanding issue. One cannot take the traditional approaches via Dedikind cuts or equivalence classes of Cauchy sequences. Even constructions via decimal expansions are problematic. There may still be epsilon-delta sorts of limits.

One may still make the distinction between rational and irrational numbers via the proof given by Hippasus of Metapontos in Pythagorean times provided one accepts LEM.

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Is that so? Well then, what does the competition have to say about this? What is the Post-Modernist, Marxist, Existentialist or Analytic-Linguistic position on cardinal infinities?Rhetorical question. They have nothing to say about it either because it is not a philosophical question.As far as the question of finding what the idea of infinity reduces to, I think that has already been answered. Infinity reduces to the open-ended nature of concepts.

Au contraire, mon ami. Mathematics is the cult of truth, putting it squarely in the domain of philosophy. What is more, Oism makes claims about infinity. Since infinity is a mathematical concept, this must be addressed from both a mathematical and philosophical point of view. If other philosophies make no claims about the infinite, then so much the worse for them.

I think we have already made headway in discussing this issue. We have identified a mathematical axiom that is objectionable.

Let me add that if natural numbers are observable properties of physical objects, then we have no objection to finite natural numbers. We have trouble talking about them as contained in one set since no such set corresponds to anything observable. Reduction has consequences. The Infinity Axiom must go.

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I think we may conclude that Oists philosophy is inconsistent with the Infinity Axiom since that axiom implies the actual existence of an infinity.

Actual in this context means that infinity is valid in a metaphysical way. There is no actual infinity any more than there is actual justice. There is a potential infinity in the sense that while at this point in time there is a definite amount, but abstractly, some patterns go on forever. I understand you're saying natural numbers are observable, and that's fine, but a *set* of numbers is necessarily abstracted from a natural number. So an infinite set is no problem. Oist philosophy is not inconsistent with the axiom of infinity. See post #91.

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Actual in this context means that infinity is valid in a metaphysical way. There is no actual infinity any more than there is actual justice. There is a potential infinity in the sense that while at this point in time there is a definite amount, but abstractly, some patterns go on forever. I understand you're saying natural numbers are observable, and that's fine, but a *set* of numbers is necessarily abstracted from a natural number. So an infinite set is no problem. Oist philosophy is not inconsistent with the axiom of infinity. See post #91.

You are confusing the concept of unboundedness with the concept of infinity.there is no bound to the Natural Numbers. That does not give you license to talk about a set of all of them. Such a collection is not reducible to perceivable reality. Again, reduction has consequences.

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I'm no mathematician, but how can any infinity be quantifiable?

There is a difference between mathematics and philosophy , yes?

Indeed there is.  Mathematics works and produces useful results.

 

As to infinity,  the precise definition of an infinite (or better still a transfinite) cardinal number is NOT  humongously so big that  you wouldn't believe how big it is!  It goes like this.   A set is infinite if it is non-empty and it can put into one to one correspondence with a proper subset of itself.

 

By this definition,  the set of integers  is infinite because it can be put into one to one correspondence with the set of even integers,  a proper subset.

 

Here is the correspondence:   n <-> 2*n.  On of the consequences of this is that there is no maximal integer.  Given any integer n there is always one larger than it,  n + 1. 

 

ruveyn1

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You are confusing the concept of unboundedness with the concept of infinity.there is no bound to the Natural Numbers. That does not give you license to talk about a set of all of them. Such a collection is not reducible to perceivable reality. Again, reduction has consequences.

Please correct me then: is infinity a concept that refers to any pattern that goes on endlessly and also refers to the whole pattern at once? That is certainly reducible (in the Objectivist sense) to perceivable reality.

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Please correct me then: is infinity a concept that refers to any pattern that goes on endlessly and also refers to the whole pattern at once? That is certainly reducible (in the Objectivist sense) to perceivable reality.

The Infinity Axiom asserts the existence of an infinite set. This is exactly analogous to my assertion that there are flying reptiles that breathe fire. You can't observe either so neither exists. This is not to say that there aren't a potential infinity of Natural Numbers. You have that without the Infinity Axiom. You may prove the Natural Numbers are not finite by reductio ad absurdum without referring to the set of Natural Numbers as a whole. Your pattern works without the Infinity Axiom.

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The Infinity Axiom asserts the existence of an infinite set. This is exactly analogous to my assertion that there are flying reptiles that breathe fire. You can't observe either so neither exists. This is not to say that there aren't a potential infinity of Natural Numbers. You have that without the Infinity Axiom. You may prove the Natural Numbers are not finite by reductio ad absurdum without referring to the set of Natural Numbers as a whole. Your pattern works without the Infinity Axiom.

Dragons aren't reducible to perception, but infinite sets are. Reducible doesn't mean perceivable, it just means a concept can be connected to reality by figuring out how the concept was developed. There's nothing problematic about infinite sets, at least nothing problematic suggested by what was discussed about the nature of concepts.

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