Irrational numbers and Physical constants

[bluecherry]

"the logical deduction of the existence of these infinities descends from LEM"

Again, infinity - any infinity - doesn't exist in reality like a cow or a rock or a star or you or me. The ability to think something up doesn't mean it must exist like the cow and such. Note that there are no one-eyed, one-horned, flying, purple people eaters for example. While infinity and the purple people eater are both things we can think of via combining aspects of various real existents in our minds and/or ignoring parts of real existents, they don't exist like the cow/rock/star/you/me. LEM doesn't lead to the conclusion that unequal infinities exist because the premise that ANY infinities exist out and about in reality like you/me/cow/etc. in the first place is wrong. Debating over whether or not there can be different size infinities is as applicable to the real world as asking if somebody could pick up Thor when Thor is holding his hammer since supposedly only Thor can pick his hammer up. This doesn't have anything to do with floating abstractions though. Floating abstractions are not things we can imagine, but which do not actually exist. Floating abstractions are pulling a concept out of the context it was formed in and attempting to apply it outside that context, like dropping the context of free-willed creatures or even living things and asking if a rock has freedom.

[Grames]

To you and Grames I would like to point out that while it is apparent that the existence of different cardinal infinities is playing with floating abstractions, the logical deduction of the existence of these infinities descends from LEM.

All abstractions do not exist, but what makes a floating abstraction is that there is no meaning in the Randian sense. 'Meaning in the Randian sense' is reference. There is a perfectly comprehensible referent for 'different cardinal infinities' which is itself an abstraction, not an existent. I disagree that different cardinal infinities are a floating abstraction anymore than complex numbers are.

[dream_weaver]

From lecture 1 of "Introduction to Logic" @ 47:30 minutes:

All laws of logic are reformulations of the law of identity.

Law of excluded middle:

Everything is either A or non-A, at a given time and in a given respect.

Between A and non-A the middle is out, there is no middle, the middle is excluded. Your choice is exclusively in the extremes. It is, or it is not. No third middle of the road possibility exists.

It goes beyond "All LEM really says is that there is no third option for certain concepts." as far as the given examples go.

ie: A man is either a genius, or he is not, at a given time (he may yet become one) and in a given respect (a genius in physics, but not in psychology.)

[aleph_1]

. This doesn't have anything to do with floating abstractions though. Floating abstractions are not things we can imagine, but which do not actually exist. Floating abstractions are pulling a concept out of the context it was formed in and attempting to apply it outside that context, like dropping the context of free-willed creatures or even living things and asking if a rock has freedom.

Is not the failure to properly contextualize a concept called a "stolen concept"? Isn't a floating abstraction a concept with no perceptual validation? I know that Peikoff was less than clear on these ideas. Isn't a dragon a floating abstraction? A dragon is a flying reptile that breathes fire. isn't creation science a stolen concept?

To Grames, would you reject an axiom that implies logically the existence of God? Surely, any theory from which we can deduce what cannot be perceptually validated should be rejected! I may be overstating the case since set theory has more than one axiom, but the set axioms and the axioms of logic imply that there are distinct infinities. The foundation of these concepts must contain meaninglessness and should be rightfully expunged from one's system of concepts. There is surely something fishy at the foundation of number!

There are additional reasons for rejecting LEM, but surely it being implicated in the production of meaninglessness makes it suspect.

[Eiuol]

There are additional reasons for rejecting LEM, but surely it being implicated in the production of meaninglessness makes it suspect.

Yes, but why must LEM contain meaninglessness? Why is having distinct infinities a problem any more than complex numbers, as Grames suggests? You keep saying there is an issue with this, but not why it is an issue for LEM. What is the meaninglessness you are referring to?

Edited February 6, 2013 by Eiuol

[bluecherry]

"The 'stolen concept' fallacy, first identified by Ayn Rand, is the fallacy of using a concept while denying the validity of its genetic roots, i.e., of an earlier concept(s) on which it logically depends."

“Philosophical Detection,”

Philosophy: Who Needs It, 22

http://aynrandlexicon.com/lexicon/stolen_concept,_fallacy_of.html

". . . would you reject an axiom that implies logically the existence of God?"

But there is no such axiom. Somebody may try to claim something like the existence of a god being axiomatic, but that's not an axiom, that's just an assertion. An axiom is something that must be true for one to even be able to try to argue against its existence in the first place. (I don't recall if LEM is actually considered an axiom though. It may be a corollary of one of them or something like that. Not that important to the discussion though.) Would one reject such a hypothetical, not actually existing axiom that said a god must exist? The question itself has nothing to do with reality, no bearing on it since the hypothetical axiom doesn't exist (and couldn't, a god is a concept that contains contradictions) just like (any) infinity doesn't actually exist. When one stops talking about stuff that actually exists then that stuff is not bound by the rules of reality and you can easily get into all kinds of nonsensical conclusions that would never be possible if you were talking about real stuff.

[aleph_1]

"The 'stolen concept' fallacy, first identified by Ayn Rand, is the fallacy of using a concept while denying the validity of its genetic roots, i.e., of an earlier concept(s) on which it logically depends."

I took this from the wiki.

A floating abstraction is a concept or idea which is, in your mind, cut off from reality, i.e., which you have not reduced to its referents.  It stands in your mind as a string of words disconnected from concretes.

". . . would you reject an axiom that implies logically the existence of God?"

But there is no such axiom.

Also,

"The “stolen concept” fallacy, first identified by Ayn Rand, is the fallacy of using a concept while denying the validity of its genetic roots, i.e., of an earlier concept(s) on which it logically depends."

The way I read these, a stolen concept is an otherwise valid concept that has been taken bereft of it's context. On the other hand, you will read in many places that 'god' is a floating abstraction, and there is no way to argue the validity of that concept. It is not stolen. It is disconnected from reality.

Now, the ideas we are talking about here are relevant since rejection of LEM results in the inability to distinguish between rational and irrational numbers. This is exactly pertinent to the original post.

I was reasoning by analogy when I referred to an axiom from which you can logically deduce the existence of god. If you have come to believe something is an axiom, from which you can deduce something you know is invalid, then you may deduce that you were mistaken about the "axiom". That is where I am placing LEM, perhaps without justification.

Concerning complex numbers, they are either just as valid as the number 2, or they are the result of the field extension of the real numbers by the inclusion of the abstraction called i. If this abstraction has no correspondence with existence, then it is invalid. The distinction between infinities is different, resulting from LEM. They are philosophically important since their existence seems to invalidate an axion of philosophy.

Finally, the books in Atlas Shrugged are named after some axioms of logic. One is Non-Contradiction. This is the Law of Dichotomy. One is Either-Or. This is LEM. The last is A is A. This is the Law of Identity. LEM is important, being one of the principle axioms of logic.

[Eiuol]

I was reasoning by analogy when I referred to an axiom from which you can logically deduce the existence of god. If you have come to believe something is an axiom, from which you can deduce something you know is invalid, then you may deduce that you were mistaken about the "axiom". That is where I am placing LEM, perhaps without justification.

Indeed, so why don't you provide a justification then? I may be a broken record here, but I really am curious.

[bluecherry]

I took this from the wiki.

"A floating abstraction is a concept or idea which is, in your mind, cut off from reality, i.e., which you have not reduced to its referents. It stands in your mind as a string of words disconnected from concretes."

Where in the wiki is that from exactly? Unfortunately, the lexicon website doesn't have a page on floating abstractions. I looked into the wiki, but this is all that is there on the page titled "Floating Abstractions": "The fallacy of the 'floating abstraction' is Ayn Rand's term for concepts detached from existents, concepts that a person takes over from other men without knowing what specific units the concepts denote." It doesn't mention any particular source there though either.

http://wiki.objectivismonline.net/Floating_Abstraction

Floating abstractions generally are concepts that do refer to actual things in reality, but which somebody else started trying to use without looking into the source of the concept and learning about the actual things the concept refers to and thus the result is that person applying the concept sloppily all over the place to things it is inapplicable to. Hence my earlier example of somebody trying to ask if a rock has freedom - freedom does refer to something in reality, but freedom has nothing to do with rocks. Angels on the other hand may be an idea somebody has without attaching it to any real thing out in the world, but that's not because they've just tried to hitch a ride on the mental efforts of others, it's because there is nothing which exists for it to refer to.

"The way I read these, a stolen concept is an otherwise valid concept that has been taken bereft of it's context."

Nein. Stolen concept fallacy is committed when somebody does something like, say, tries to argue that nothing exists when arguing wouldn't be possible if nothing existed. Or somebody who tries to say that mirages prove the senses are unreliable when they relied upon sensory evidence to learn of mirages. Stolen concept is trying to use a concept A, counting on it to be true, in attempting to claim that concept A (or another concept B that is necessary to derive concept A) is false.

"I was reasoning by analogy when I referred to an axiom from which you can logically deduce the existence of god."

I'm aware it was an analogy. Analogies aren't arguments though, they are just a way to try to help clarify to somebody what your actual argument means. The fact that there is no such god-inducing axiom is something in common with infinity - neither exists and the fact that both of these are not part of reality means they don't necessarily follow the rules of reality. LEM is a rule of reality. The ability to think up non-existent stuff and find that non-existent stuff doesn't obey the rules of reality can tell one nothing about how reality works aside from maybe a bit about the way the human mind can function because reality and non-existent stuff have as close to nothing to do with each other as could be. You can't invalidate LEM, a rule of the functioning of reality, by reference to the functioning of infinity, a non-reality.

[GrandMinnow]

There are some inaccuracies that have been stated in this thread about Cantor's theorem (and other theorems) with regard to the law of excluded middle.

Note: The following is an account of certain basics in set theory and mathematical logic. These points might not be accepted in Objectivism, but they are correct in the context of Cantor and set theory, as these have become a subject in this discussion. The terminology used here is with regard to set theory and mathematical logic and is not necessarily in accord with Objectivist terminology, but one cannot discuss these matters about Cantor and set theory that have been raised in this thread without keeping clear terminology within the context of set theory and mathematical logic itself.

Cantor's theorem was originally proven by Cantor in his set theory that was not formalized at that time. Subsequently, set theory has been formalized in a number of ways, most commonly in certain variations of Z set theory such as Z, ZF, and ZFC or in class theories such as NBG. And Cantor's theorem may be formally proven in those formalizations also.

Cantor's theorem is that there is no surjection ('surjection', 'injection' and 'bijection' are not Cantor's own terminology, but they work just as well in this context) from a set S onto the power set of S (thus, there is no bijection (i.e. 1-1 correspondence) between a set and its power set). Cantor proved similar theorems, such as that there is no surjection from the set of natural numbers onto the set of denumerable binary sequences (thus no surjection from the set of natural numbers onto the set of real numbers).

These proofs do NOT rely on the law of excluded middle and these proofs do NOT rely on non-intuitionistic proof by contradiction. The logic used for these proofs is intuitionistically and constructively valid.

There are two forms of proof by contradiction. One is intuitionistically valid and does not require the law of excluded middle, while the other is not intuitionistically valid and does rely on the law of excluded middle:

(1) Assume P. Show a contradiction. Infer the negation of P. This is intuitionistically valid and does not use the law of excluded middle.

The sentential analogue of this is: If P then [something false], therefore P is false. This is intuitionistically valid and does not use the law of excluded middle.

(2) Assume the negation of P. Show a contradiction. Infer P. This is not intuitionistically valid and does use the law of excluded middle.

The sentential analogue of this is: If the negation of P then [something false], therefore P is true. This is not intuitionistically valid and does use the law of excluded middle.

Cantor's proofs, if we couch them as proofs by contradiction, may be given as form (1) above, which is intuitionistically valid and does not use the law of excluded middle.

Cantor's proofs, of course, rely not only on the logic used but on certain premises. However, the minimal premises used to prove Cantor's theorem are (as far as I know) acceptable both for intuitionists and predicativists. (Even though the axiom schema of separation may not be in general acceptable to the predicativist, the particular instance used in proof of Cantor's theorem is, as far as I know, acceptable to the predicativist.)

However, there are further conclusions drawn, such that there are infinite sets that are not in bijection and that there are "greater and greater infinite sets" (i.e., that for any infinite set S there is an infinite set T such that S injects into T but S does not surject onto T (thus there is no 1-1 correspondence between S and T). These conclusions do rely on certain premises (general principles if informal; axioms if formal), such as the power set axiom and the axiom of infinity that the intuitionist may reject. But these conclusions do NOT rely on the law of excluded middle.

Edited February 7, 2013 by GrandMinnow

[GrandMinnow]

Additional points in response to certain mentions here:

(1) As well as the law of excluded middle not being necessary to prove the various theorems mentioned, it is not the case that the law of excluded middle is sufficient to prove the various theorems mentioned. Those theorems are proven from certain set theoretic principles or axioms and the theorems are not provable merely by application of the law of excluded middle or even application of the law of excluded middle along with other logical principles or axioms. It is NOT the case that

"The existence of distinct infinite cardinalities is a logical consequence of LEM."

(2) The reason Brouwer and the intuitionists rejected the law of excluded middle is not Cantor's theorem or the various theorems about greater and greater infinite sets. As mentioned, the law of excluded middle is not required for those results. Rather, the reason Brouwer and the intuitionists rejected the law of excluded middle is that it is non-constructive in the sense that it allows proving that there exists an object with a certain property but without also pointing out any specific such object. A constructive proof is one such that if it proves that there exists an object with a certain property, in then the proof itself may be used to further specify a particular object that has the property. By the way, intuitionists do not reject the law of excluded middle if it is used only for arguments pertaining to finite domains. If only finite domains are countenanced, then intuitionists DO accept the law of excluded middle, since with finite domains, a finite search will produce a specific object having a certain property if there does exist such an object. It is only where infinite domains are allowed that the intuitionist rejects the law of excluded middle.

(3) Points usually do not have dimensions (size or distance in different dimensions) themselves with regard to the system of dimensions in which the points are points. However, that does not make points "nothings". Points are mathematical objects (whether they be pairs of real numbers or merely unspecified as geometric points in a given geometry, or whatever, per the system considered) and have properties. That in the system the points do not themselves have distances within them (since usually distance is a function of pairs of points, not a function of a point onto itself) does not vitiate points so that they are "nothings".

[Grames]

Thank you GrandMinnow for that contribution. But just to clear up my part in this:

Points are mathematical objects, therefore they are "nothings" precisely because a mathematical object is an abstraction and not an existent. This is not an attack on the concept of a point. Points are valid tools for thought, and they need not be existing things to be valid tools of thought, because they are (or can be considered to be) attributes abstracted from existents. (A point is 'related to/formed from' a real existing thing by omitting every other attribute except a location.)

Now if you or anyone wants to claim that points are real existing things rather mathematical objects then I would have a problem with that claim.

[GrandMinnow]

Points are mathematical objects, therefore they are "nothings" precisely because a mathematical object is an abstraction and not an existent.

I don't opine as to the Objectivist notions here (though, do I recall incorrectly that with Objectivism, abstractions also are existents (or is it "entities"?), indeed that to deny that abstractions (such as concepts?) are existents (or should I say "entities"?) is to suffer from being "concrete bound"?).

But what I am saying is merely in context of ordinary mathematics in which points are objects as much as any other mathematical object. For example, the point in the real plane <0 1> is no less an object than the number 2 or the unit circle or any other mathematical object. So, if a point is a "nothing" than so are other mathematical objects such as the number 2 or the unit circle. Indeed, a line itself is a certain kind of set of points, and thus an abstraction, and if points are "nothings" than a line too is a "nothing" and indeed it is "nothing" set of points that are themselves "nothings".

Actually, I don't even care to press a case as to "objecthood" (I don't care to press the case for any particular ontological notion of mathematics) EXCEPT the sense that IF points are "nothings" then so are such "things" (whatever their ontological or even merely linguistic "figure of speech" status, or possibly in Objectivism "concepts of method"?) as numbers, lines, and circles, complex numbers, the complex system itself, etc., especially given that in, for example, the real plane, a point is merely an ordered pair of real numbers. Particularly, that a point itself doesn't have a distance (along one of the two dimensions) is not good grounds for saying that a point is a "nothing". To illustrate (but not to argue) by analogy: Age difference is a relation between people and is not a relation a person has onto himself. It makes sense to ask "what is the age difference between Mike and Jim?" but makes no sense to ask "What is Jim's age difference?". But we don't infer then that Jim is a "nothing". It makes sense to ask, "what is the distance between the point <2 4> and the point <7 15>?" but no sense to ask "what is the distance of the point <7 15>?". Yet it seems, at best, odd to then infer that the point <7 15> is a "nothing".

[Grames]

(I don't care to press the case for any particular ontological notion of mathematics)

Ok, we can stop here then.

[Plasmatic]

Minnow, concepts are mental existents but even Ms. Rand used the same type of language that caused your query. Grames means that abstractions are not mind independent existents.

[ruveyn1]

The mathematical point (an abstraction) is pure place, pure location. It does not have any kind of extension. In the original greek Euclid's "definition" of a point translates as "a point is that which has no parts". Σημεῖόν ἐστιν, οὗ μέρος οὐθέν

See: http://farside.ph.utexas.edu/euclid/Elements.pdf

The interesting thing is that this "definition" is never used in a proof. Not once in the 13 books of Euclid. It is the postulates that make assertions about points that essentially define the points.

Ditto for Euclid's "definition" of a line: A line is a length without breadth. Γρaμμὴ δὲ μῆκος ἀπλaτές

But the only thing we "know" about lines is stated in the postulates.

Postulate 1. To draw a straight line from any point to any point.

Postulate 2. To produce a finite straight line continuously in a straight line.

Postulate 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. and so on. ruveyn1

[ruveyn1]

The mathematical point (an abstraction) is pure place, pure location. It does not have any kind of extension. In the original greek Euclid's "definition" of a point translates as "a point is that which has no parts". Σημεῖόν ἐστιν, οὗ μέρος οὐθέν

See: http://farside.ph.ut...id/Elements.pdf

The interesting thing is that this "definition" is never used in a proof. Not once in the 13 books of Euclid. It is the postulates that make assertions about points that essentially define the points.

Ditto for Euclid's "definition" of a line: A line is a length without breadth. Γρaμμὴ δὲ μῆκος ἀπλaτές

But the only thing we "know" about lines is stated in the postulates.

Postulate 1. To draw a straight line from any point to any point.

Postulate 2. To produce a finite straight line continuously in a straight line.

Postulate 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

and so on.

ruveyn1

[GrandMinnow]

concepts are mental existents but [...] abstractions are not mind independent existents.

Yes, that is my understanding of Objectivism also. Edited February 8, 2013 by GrandMinnow

[aleph_1]

I would like to reply to Grand Minnow, but my iPad does not permit me to quote easily.

First, the Stanford Encyclopedia of Philosophy states that Intuitionistic Logic is succinctly, logic without the law of the excluded middle. LEM is also equivalent to the Law of Double Negation Exclusion. This mean that one may not replace not(not(A)) with A in a statement. However, A can be replaced with not(not(A)) in Intuitionistic proofs.

That being said, Cantor's Theorem (CT)

Card(A)<card(P(A)).

The proof goes as follows: Assume not(CT). Deduce not(not(CT)). Therefore CT. This is precisely the form of proof rejected by Intuitionists.

However, this is not the point of this post. Since objectivists reject concepts that have no reduction, it follows that objectivists must reject as philosophically invalid the Infinity Axiom, and hence must reject the classical construction of the Natural Numbers. This puts objectivists in the camp of the ultrafinitists, or at least the strict finitists. Any mention of "potential infinities" is a reference to a concept equivalent to god. This also means that Objectivism has difficulties with Real Numbers, limits, calculus, etc.

Finally, I would like a good reference that clearly explains the objectivist definitions of floating abstractions and stolen concepts. You see all sorts of things on the web that contradict bluecherry's definitions.

[GrandMinnow]

In most succinct terms, your claim that an Objectivist's acceptance of the law of excluded middle demands acceptance of greater and greater infinities is incorrect.

 

First, the technical matter:

 

Here is the ordinary proof of Cantor's theorem:

 

['P' for 'power set of' and '~ ' for 'it is not the case that'.]

 

Claim (Cantor's theorem): For any set S, there is no function from S onto PS (i.e. there is no surjection from S onto PS).

 

Proof: Let f be any function from S into PS. Let D = {x | x in S & ~ x in f(x)}. Suppose D is in the range of f. Then, for some x in S, we have f(x) = D. But if x is in D then x is not in D, and if x is not in D then x is in D, a contradiction. Therefore, there is no x such that f(x)=D, i.e., D is not in the range of f. But D is in PS. So f is not onto PS. Since f was an arbitrary function from S into PS, there is no function from S onto PS.

 

No use there is made of the law of excluded middle or double negation or intuitionistically invalid proof by contradiction. Rather, the intutitionistically valid form of proof by contradiction was used.

 

And of course, it follows that card(S) < card(PS), since there is the obvious injection from S into PS but (as just proved) no surjection from S onto PS (perforce no bijection between S and PS). (Recall that card(S) < card(PS) if and only if there is an injection from S into PS but no bijection between S and PS.)

 

In the proof, we let P stand for "D is in the range of f", then the argument is of the form:

 

Suppose P. Then a contradiction. Therefore ~P.

 

And that is intuitionistically valid. 

 

Or, suppose we wish to put it in terms of card from the start:

 

Claim: card(S) < card(PS).

 

Proof: Recall that cad(S) < card(PS) if and only if there is an injection from S into PS and there is no bijection between S and PS. Now, there is the obvious injection from S into PS. So we need only show that there is no bijection between S and PS. By the proof above in this post, there is no surjection from S onto PS. Therefore there is no bijection between S and PS. Therefore, card(S) < card(PS).

 

Again, intuitionistically valid. 

 

aleph_1, on 08 Feb 2013 - 10:37: "However, this is not the point of this post. Since objectivists reject concepts that have no reduction, it follows that objectivists must reject as philosophically invalid the Infinity Axiom"

 

I don't contest one way or another that Objectivists may wish or need to reject the axiom of infinity. More fundamentally, Objectivists may reject even the very framework of set theory itself so that the axiom of infinity would be rejected perforce along with the entire set theoretic framework (but that is for Objectivists to decide).

 

What was incorrect in your previous claims is that to avoid that there are greater and greater infinities or Cantor's theorem, et. al, Objectivists (or anyone) would have to reject the law of excluded middle, and incorrect is your claim that the law of excluded middle logically demands Cantor's theorem or that there are greater and greater infinities. An Objectivist (or anyone) may quite happily and with perfect consistency accept the law of excluded middle while rejecting Cantor's theorem or that there are greater and greater infinities. To reject Cantor's theorem while still holding to the law of excluded middle can be done by, for example, (1) rejecting the very set theoretic framework of Cantor and his more formal successors, (2) rejecting the premises of Cantor's argument, such as the axiom schema of separation (not an axiom schema of Cantor himself, but an instance is used when we couch Cantor's argument in a formal axiomatic theory). And to reject that there are greater and greater infinities while still holding to the law of excluded middle can be done by, for example, (1) rejecting the very set theoretic framework of Cantor and his more formal successors, (2) (as you mentioned) rejecting the axiom of infinity, or rejecting the power set axiom.

 

In most succinct terms, your claim that an Objectivist's acceptance of the law of excluded middle demands acceptance of greater and greater infinities is incorrect.

 

aleph_1, on 08 Feb 2013 - 10:37: "Any mention of "potential infinities" is a reference to a concept equivalent to god." 

 

I think you meant mention of 'actual infinity'? If I recall correctly, Objectivism, per Ayn Rand herself, does accept the notion of potential infinity while, of course, rejecting actual infinity.

Edited February 8, 2013 by GrandMinnow

[Eiuol]

In most succinct terms, your claim that an Objectivist's acceptance of the law of excluded middle demands acceptance of greater and greater infinities is incorrect.

Since you are certainly willing to explain yourself and it has helped me understand, what is in fact wrong with accepting greater and greater infinities in the first place? What I'm thinking is that while some patterns can go on infinitely, the magnitude in which patterns go on infinitely can vary, which seems to suggest that "greater infinities" makes sense. Some magnitudes can be greater than others depending upon your level of abstraction.

Edited February 8, 2013 by Eiuol

[GrandMinnow]

 what is in fact wrong with accepting greater and greater infinities in the first place?

I have no objection to there being sets of greater and greater infinite cardinality. But, as far as I can tell, Objectivists tend not to look kindly on the notion. First of all, Objectivism rejects that there are actually infinite sets, let alone actually infinite sets of greater and greater cardinality. But even more fundamentally, it's likely that an Objectivist notion of 'set' differs from how set theorists regard the notion of 'set'.

 

(On the other hand, Steven Simpson is an Objectivist and a very prominent mathematician and logician who works with classical mathematics (which includes set theory with its actually infinite sets and sets of greater and greater infinite cardinality). From my oral conversation with him, it's not clear to me how he reconciles Objectivism with set theory; and indeed he mentioned that he thinks there is work to be done to produce a more robust Objectivist account of mathematics.)

Edited February 8, 2013 by GrandMinnow

[Grames]

First of all, Objectivism rejects that there are actually infinite sets,

 

It would more accurate to state that Objectivism rejects that there are sets.   Sets are epistemic artifacts, created by regarding some things as similar and others not similar.  Infinities can also validly be conceived as epistemic artifacts.  Neither exists as an actual, or in other words neither are valid in metaphysics.

 

 

Edited February 8, 2013 by Grames

[Eiuol]

 

It would more accurate to state that Objectivism rejects that there are sets.   Sets are epistemic artifacts, created by regarding some things as similar and others not similar.  Infinities can also validly be conceived as epistemic artifacts.  Neither exists as an actual, or in other words neither are valid in metaphysics.

 

 

I certainly agree that infinity doesn't exist in a metaphysical sense any more than concepts in general exist in a metaphysical sense. Insofar as they are epistemological though, sets seem quite important for epistemology. If my math terminology is wrong here, anyone can correct me. I could call all objects that fall under the concept apple as Set{A}, and everything else as Set{~A}. Obviously Set{A} is things that are similar, and Set{~A} is things that are not similar to Set{A}. In other words, Set{~A} is just the set of everything else outside of Set{A}. Not only that but concepts refer to past, present, and future, so Set{A} can grow and change infinitely. I fail to see where sets are even a possible issue for Objectivist epistemology if concepts seem to behave as any set possibly could. I fail to see an issue with infinite sets, too, because, well, concepts are sets that go on forever in both time and number of referents, used for an epistemologically purposeful end.

Edited February 9, 2013 by Eiuol

[Grames]

Exactly Eiuol.  Concepts are open-ended in the number of possible referents, which is suspiciously similar to the idea of an indefinitely large set.