GrandMinnow

I understand. I would expect that from an Objectivist point of view.

I don't know what specific writers you have in mind when you mention "the empiricist program". In whatever sense Hilbert's views can be described in terms of empiricism, perhaps the most salient aspect of his view was in the distinction between the contentual (which is finitistic) and the ideal (such as abstract set theory with infinite sets) and the proposal to use only finitistic mathematics to prove the consistency of ideal mathematics. Russell though is famous for his own version of empiricism in general and logicism in mathematics. Principia Mathematica (PM) has not been vitiated by any paradoxes, as far as I know. Rather, it was Frege's system that was shown to be inconsistent by Russell, and PM was devised by Whitehead and Russell as a remedy. Then Godel showed PM to be incomplete, but he did not show it to be inconsistent (anyway, trivially, any inconsistent theory is complete). Logicism is another matter. Whatever we may say of logicism, it is commonly asserted that PM itself does not do the job since PM uses three axioms that one would be inclined to say are not logical: infinity, choice, and reducibility, as even admitted by Russell. However, even on that point there is debate on technicalities and some informed holdouts who propose their own versions of logicism.

I don't have a firm view on the matter. But I think it is safe to say that if incompleteness weighs in at all about formalism then it does so in disfavor of formalism and it's hard to imagine that it does so in favor of formalism. But again I caution that this depends on a more precise understanding of formalism and should not depend on common misconceptions about what formalism is. In any case, in the specific sense I mentioned - Hilbert's proposal for a finitistic proof of the consistency of the ideal mathematics of set theory, of analysis, and even of arithmetic - the incompleteness theorem would seem to defeat the proposal, and as I understand, this accords with a consensus of informed writers on the subject, though there is still some debate on the technicalities and even some informed dissenters.

Then I don't follow what you meant by "implies the opposite" if not to say that Molineux's proposed refutation of formalism is rebutted, and "for example" if not to say that the independence of the continuum hypothesis is part of what is implied by incompleteness in the passage: "You say, "Mathematical formalism [...] was defeated by Kurt Godel's incompleteness proof." It seems that Godel's Incompleteness Theorem implies quite the opposite of what you are saying. For example, Godel proved that you cannot prove the Continuum Hypothesis." [alpeh_1] But if I have misunderstood you then I stand corrected, especially since my point is not to pin anyone down here but rather to keep straight the mathematics discussed and to suggest restraint in drawing philosophical conclusions based incorrect or vague understandings of the mathematics. In any case, it seems wrong to say that the incompleteness theorem does the "opposite" of refuting formalism. If it is correct to claim that the incompleteness theorem weighs in at all about formalism then it does so against formalism and not in favor of it.

Yes, my mistake.

I didn't mean to do that. My main point is that incompleteness and some other topics mentioned here are technical mathematical results, so one needs to properly study the actual mathematics before using this mathematics as a basis for philosophical claims. I don't deny that there is a rich philosophical discussion that comes in the wake of the mathematics, but for these discussions the mathematics must first be carefully understood. Then, putting the matter in Objectivist context further complicates this, since the mathematics itself is not devised to necessarily comport with the Objectivist framework.

There are several confusions in this thread about incompleteness, mathematical logic, and set theory that I comment on here. I might not have time to address responses or disputes to this post. "Mathematical formalism (the idea that math is "analytic" or just a matter of the extrapolation of concepts whose definitions we create) was defeated by Kurt Godel's incompleteness proof." [John Molineux] I don't wish to opine on the above as a description of the philosophy of formalism in mathematics, but I can say a few things. The philosophy of formalism in mathematics is usually centered on the ideas Hilbert and while I am not an expert on Hilbert's philosophy of formalism, I think I know enough to note that his view is often badly mischaracterized, especially on the Internet. Most saliently, the notion "mathematics is just a game of symbols" is often ascribed to Hilbert (though not by the poster), but there does not seem to be anything written by him that justifies characterizing his view that way. While Hilbert stressed the finitary/concrete/contentual aspect of mathematics, to my knowledge, he did not claim that mathematics does not involve mind, abstraction, or conceptualization, and whatever he might have said about analyticity, I don't know that he ever wrote anything that is fairly paraphrased as "mathematics is just a matter of extrapolation form concepts whose definitions we create"). Hilbert did famously embrace Cantorian set theory with its abstract infinite sets. What Hilbert hoped to do was prove in a finitary way the consistency of the mathematics of infinite sets (in later context, we would refer to the formal theory ZFC rather than Cantor's unformalized work itself). Arguably, the second incompleteness theorem does defeat Hilbert's hope, though it would take a more confident writer than me to opine that this is conclusive. / "Godel proved that you cannot prove the Continuum Hypothesis. Paul Cohen proved that you cannot disprove the Continuum Hypothesis." [aleph_1] That is incorrectly switched. Let us say that the consistency of ZF is taken as a tacit condition for these relative consistency matters: Godel proved that ZF does not prove the negation of the axiom of choice (i.e., he proved the relative consistency of ZFC) and that ZFC does not prove the negation of the continuum hypothesis (CH) nor of the generalized continuum hypothesis (GCH) (i.e., he proved the relative consistency of ZFC+CH). Cohen later proved that ZF does not prove the axiom of choice (i.e., he proved the relative consistency of ZF+~AC) and that ZFC does not prove CH (perforce GCH) (i.e., he proved the relative consistency of ZFC+~GCH). "You say, "Mathematical formalism (the idea that math is "analytic" or just a matter of the extrapolation of concepts whose definitions we create) was defeated by Kurt Godel's incompleteness proof." It seems that Godel's Incompleteness Theorem implies quite the opposite of what you are saying. For example, Godel proved that you cannot prove the Continuum Hypothesis." [alpeh_1] As I mentioned, that is not what Godel proved. And it is not well put to say that the relative consistency and independence of the continuum hypothesis is an implication of the incompleteness theorem. And I don't see how the relative consistency and independence proofs refute a refutation of formalism. By the way, speaking of the independence of the continuum hypothesis, proved by Cohen, we note that Cohen considered himself a formalist. "It follows that this hypothesis is beyond logic and represents pure rationalism since it is also beyond observation." [aleph_1] I don't know why we would have to go to such elaborate independence results for this, depending on what is meant by "logic". These independence results are not discussed by mathematicians in a sense of 'logic' limited to Ayn Rand's definition of 'logic'. In the context of the mathematical logic in which such independence results reside, there are theorems of (pure) logic, such as those of the mere first order predicate calculus, and there are specifically mathematical theorems that are not theorems of pure logic (they are non-logical theorems) . Even "0+0=0" is such a mathematical theorem that is not a theorem of pure logic but rather depends on non-logical mathematical axioms. Granted, theorems such as "0+0=0" are finitary, but still we can easily point out non-finitary theorems that, without such elaborate results as the independence of the continuum hypothesis, trivially we can show not to be theorems of pure logic. Whether such theorems are "rationalism" per Objectivism, I would leave for Objectivists to say. Meanwhile, without involving such elaborate results as Godel's or Cohen's we can trivially show that the axiom of infinity itself is a non-logical axiom. Arguably, this alone defeats Russell's philosophy of logicism that wishes to prove the theorems of mathematics (say, real analysis) from purely logical axioms though, it would take a more confident writer than me to opine that this is conclusive. "Godel showed that mathematics contains theorems that cannot be proved through mathematical logic, and that mathematics cannot be used to prove its own consistency." [aleph-1] By definition, in the context of the mathematical logic in which this work is done, a 'theorem' is a statement that is proven from some set of axioms. What you may be referring to is that Godel-Rosser showed that for any consistent recursive axiomatization, there are true statements about natural numbers that are not theorems from said axioms. Meanwhile, of course there are theorems of mathematics not provable by pure logic, but that is a rudimentary fact of beginning logic that does not need anything so elaborate as the incompleteness theorem. / "Do you relate the self reflexive nature of Godel.s theorom to the type of tautalogical circularity of axioms?" [Plasmatic] What "self-reflexive nature of Godel's theorem"? What tautological circularity of what axioms? Of course, the purely logical axioms are validities - either tautologies of sentential logic or validities of predicate logic. In any case Godel's incompleteness theorem itself is provable from purely finitistic assumptions: Proof of the incompleteness theorem may be shown by reasoning confined to intuitionistic logic and mathematical assumptions that do not extend past those of basic arithmetic. / "But such statements can never be justified by pure experience, per the problem of induction. And they are NOT analytic--that is precisley what Kurt Godel proved they are not (at least with mathematics)!" [John Molineux] The Godel-Rosser incompleteness theorem is (in contemporary terminology) that for any recursively axiomatized, consistent, "arithmetically-adequate" (my term for a notion that requires more technical elaboration) theory (such as Robinson arithmetic (Q) or primitive recursive arithmetic (PRA)), there are statements in the language of the theory such that neither the statement nor its negation are theorems from said axioms. And this proof may be given from merely finitistic assumptions. And, for such a theory, there are true statements of arithmetic that are not theorems of the theory. Moreover, for such theories, there is an algorithm to produce a statement such that neither it nor its negation is a theorem. Whether this suggests about analyticity, it is not part of what was PRECISELY proved, but rather requires philosophical argument from the mathematical result itself. / "Godel showed that no axiomatic system can prove itself." [Eiuol] Where does Godel show anything like that? What does it even mean to say that an "axiomatic system proves or does not prove itself"? An axiomatic system proves theorems, but an axiomatic system itself is not something that is proved. This does not require anything from Godel. Perhaps what you have in mind is that Godel's second incompleteness theorem is that no recursive, consistent, arithmetically-adequate theory can prove its own consistency. This has nothing to do with "a system proving itself" nor does it even entail that there are not axiomatic systems that don't prove their own consistency. "Godel proved that math is -only- a matter of extrapolating concepts." [Eiuol] What specific proof by Godel are you referring to? The incompleteness theorems are mathematical results. Claims such as "math is only a matter of extrapolating concepts" are philosophical. Godel did have philosophical follow-ups to his mathematical results, but I don't know what writings by Godel or in the philosophical literature you have in mind. / "[...] what about the continuum hypothesis? Are there infinite cardinalities between the countable infinity and the first uncountable infinity?" [aleph_1] No, of course there are not. It would be self-contradictory to say there is an uncountable cardinal less than the first uncountable cardinal. And that has nothing to do with the continuum hypothesis. The continuum hypothesis is that the cardinality of the set of real numbers is the first uncountable cardinal. Whatever the case may be about that, it couldn't possibly bear on the fact that, by definition, there is no uncountable cardinal less than the first uncountable cardinal. "Concerning the Continuum Hypothesis, since that is the premier application of Godel's Incompleteness Theorem [...] Standard math accepts it [...]." [aleph_1] I don't know what is meant by "the premier application". In any case, we don't need to go to the continuum hypothesis to find prominent, "substantive" and, indeed, arithmetical exemplars of the incompleteness theorem. And what "standard mathematics" accepts or depends on or even has anything to do with the continuum hypothesis? / "I'm not aware of math where 2+2=0." [John Molineux] For example, modular arithmetic for, say, a clock that has four positions 0, 1, 2, 3 and "+' for moving clockwise among positions.

I don't very much doubt that one can set up tests gaining various results. Indeed, there have been cases (or at least one case) in which children's art was bought for high prices as it was mistaken for professional painting. But this particular test (at least from what we know of it here) is not quite credible for these reasons: (1) The panel of artists and the one critic are not identified. We don't know their qualifications. (2) It is not made clear that the examples in the test here are the same as those in the test given to the original panel. (3) Overall, it is not even very hard to tell the difference anyway. This decisively negates the test. The result is the OPPOSITE of any debunking: With the particular examples in the test - it takes only a modicum of sensitivity to art to tell the difference. (4) A better test would be of full paintings seen in person and not cropped portions of image files seen on monitors of various quality. And the supposed counterexample that this would not be required for a test involving representational art is not convincing since non-representational pieces may depend more on an evaluation of their whole. If one were SINCERE to find out about viewer discernment, then one would provide reasonable context, not instead to contrive to make discernment obscured by showing only small sections (in which there is no evaluation of the success or lack of success in making an overall composition) and not to draw too firm conclusions from responses to mere image files (in which the actual techniques of brushstroke, et. al are not as apparent).

I edited my error out as soon as I realized I forgot that it was 11, not 12. This is of no consequence. P.S. And the point is not that we're "great at modern art" but rather that even a non-expert can have a good enough eye to basically see the difference, at least per this particular test.

By 'author' you mean the person who made the painting? If so, I don't see your point. My point is that a painting is evaluated as a whole painting not just a cropped portion, and best evaluated by seeing the painting itself rather than a reproduction, especially as reproduced as a computer image file viewed on this or that computer monitor.

Wait, the article says only that on a certain test some artists and critics did poorly. The article does not say that this is the SAME set of paintings as in this particular article (is it?). I'd like to see a panel of artists and critics, all named (not merely described anonymously as "artists and art historians"), and all of whom are clearly accomplished in art history and criticism, and they are shown actual paintings, not reproductions and certainly not just images seen on this or that computer monitor. And, of course, how a bunch of random people walking through a shopping mall score is pretty much irrelevant. By the way, a friend of mine just took the test and got 11/11, and like me he has no special background in painting and even less involvement in the visual arts than I have.

9/11 and they were pretty obvious. And I am not a painter nor do I have any special background in the subject of painting. Two points: (1) This would be easier if the entire canvass were shown. (2) Easier if the actual painting (rather than on a computer screen) were shown.

(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?

If I understand you correctly, your idea seems to be that without the continuum hypothesis there may be a cardinality between aleph_0 and aleph_1, which could be called "aleph_(1/2)"? Is that your idea? If so, let me explain why it is confused: (1) The aleph operation is defined by transfinite recursion on the ordinals. The alephs do not take arguments other than ordinals. Period. (2) Without the continuum hypothesis there may be a cardinality between aleph_0 and 2^(aleph_0). But without the continuum hypothesis there is still no cardinality between aleph_0 and aleph_1. By DEFINITION, aleph_1 is the least cardinal greater than aleph_0. By DEFINITION, irrespective of the continuum hypothesis, there is no cardinality between aleph_0 and aleph_1. Put another way, if we negate the continuum hypothesis, then still there is no cardinality between aleph_0 and aleph_1 (by DEFINITION, irrespective of the continuum hypothesis, aleph_1 is the least cardinal greater than aleph_0); rather, it's just that when we negate the continuum hypothesis it is not the case that aleph_1=2^(aleph_0). P.S. ZFC does not include the continuum hypothesis, and ZFC does not prove one way or the whether aleph_1=2^(aleph_0). But still ZFC does prove that there is no cardinality between aleph_0 and aleph_1, as that is true by DEFINITION.

By the way, my all-caps are not meant as shouting, but rather as tantamount to italics (without using special coding).

"The generlized continuum hypothesis is that if in is a cardinal number, then there is no cardinal number between n and 2^n." Yes, where n is infinite. "We ordinarily take aleph1 to be 2^aleph0." No, we do not. To say that aleph_1 = 2^(aleph_0) is to adopt the continuum hypothesis. But the continuum hypotheis is independent of ZFC. Without adopting the continuum hypothesis, ZFC is agnostic as to the question whether aleph_1 = 2^(aleph_0). "It is a mere hypothesis that there is no cardinal number between." All the axioms are hypotheses. And ZFC does NOT include the continuum hypothesis as an axiom. "You may, at your whim, take such an intermediate to be aleph one-half." No, you may not COHERENTLY do so. The alephs are by DEFINITION an operation on the ordinals. To mention an aleph subscripted by anything other than an ordinal is nonsense and indicates that the person does not know what the alephs are. If one wishes to define some OTHER operation that takes arguments that include non-ordinals then fine, one can go ahead and state the definition, but then it is something other than the aleph operation. "I do not think that this is a misunderstanding." It is a FUNDAMENTAL misunderstanding. "I also do not think it inappropriate to ask a kid to demonstrate some understanding of cardinality before taking his question seriously." I don't think it's inappropriate to ask him anything about mathematics. I'm just pointing out that, whether the person is ten days old or ten millennia old, the uncountabilty of the reals does not bear upon explaining why the expression "aleph_(one-half)" is a fundamental confusion. "Showing the uncountability of the reals at minimum shows that there are distinct notions of infinity." Not different notions of 'is infinite' but rather that there are infinite sets of different cardinality. Anyway, it doesn't take the example of the reals to achieve this result. More simply: By an easy proof, every set is dominated by its power set. Therefore, the power set of the set of natural numbers is uncountable. "My claim is that theories that split infinities contain meaninglessness. Is this not relevant?" I don't begrudge you from holding that set theory is meaningless. I'm just pointing out that the answer to "is there an aleph_(one-half)?" is no, there is not, by the very definition of the aleph operation, and I'm pointing out that the uncountability of the reals (and now in the discussion) the continuum hypothesis is extraneous to the explanation.

The answer to that question is no, by definition, there is no aleph_one-half. Asking whether there is an aleph_one-half shows a very basic lack of understanding of what the alephs are. Showing the uncountability of the reals has nothing to with it. (1) It can be argued that the axioms are not merely whimsical and that, as I just mentioned in my previous post, the axioms reflect our (editorial 'our') basic notion of "set". (2) Moreover, the axiomatic method does not need to be understood as making flat unqualified assertions but rather as providing a set of assertions that are true only of certain structures. (3) As I've been saying, it's not clear that Objectivism endorses even the notion of formal axiomatics. If one does not seek a formal axiomatization of the mathematics for the sciences then one does not even have any skin in the game of trying to get axioms that provide the mathematics for the sciences while not overshooting. Back to the high jumper analogy regarding this. And of course I'm not arguing that anything is settled by majority opinion. Rather, as you can see from the context of my posts, I'm only pointing out the fact that the goal of an axiomatization of the mathematics for the sciences is acheived by set theory faithful to mathematicians' notion of "set". I'm not arguing that that mere consensus makes set theory otherwise "meaningful", correct, or whatever.

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Also just as important to mathematicians is that the axioms are in keeping with our basic notions of sets. Consider the axioms of full ZFC [all the variables mentioned here range over sets]: (1) Extensionality. Sets are determined entirely by their members. That is, X and Y are equal if and only if every member of X is a member of Y and vice versa. (2) Union. For any X, there is the union of X. That is, for any X there is the set whose members are exactly those that are a member of some member of X. (3) Power set. For any X, there is the power set of X. That is, for any X there is the set whose members are exactly those that are subsets of X. (4) Schema of Replacement. For any X and any functional property from X, there is the set whose members are exactly those that are the correlated set by the functional property of some member of X. That is, for any X and any correlation from X, there is the set whose members are exactly the sets that are correlated to. (5) Infinity. There is a set that has the empty set in it and has in it the successor of any member. (The empty set will have been defined and proven to exist by axioms (4) and (1). And the unordered pair operation and singleton operations while have been defined from axioms (3) and (4). And binary union will have been defined by axiom (2) and the unordered pair operation. So that 'successor' is defined by 'the successor of n = the binary union of n with the singleton whose only member is n'.) (6) Choice. For any X that does not have the empty set has a member, there is a set "made" by "collecting" from X exactly one member of each member of X. That is, there is a choice function for every set that does not have the empty set as a member. (7) Regularity. For any nonempty X, there is a member y of X such that no member of X is a member of y. That is, every set has a membership-minimal element. This entails that sets are not members of themselves and don't have infinitely descending membership sequences, etc. / It's a pretty safe bet that the vast majority (there are some dissenters) of mathematicians (who are even concerned about such foundational matters) find these principles to be faithful to our notion of sets.

I don't know. However, consider that the set of real numbers is an uncountable set. And the real numbers and the calculus of the real numbers is the starting context for mathematics for the physical sciences. But it turns out that while the set of real numbers is uncountable, our set theory does not determine what aleph is the cardinality of the set of real numbers (this is the famous continuum problem). In other words, the theory proves that the cardinality of the set of real numbers is an aleph, but the theory does not prove which aleph it is. Also, consider the context here. Suppose we wish to have an axiom system (I don't mean 'axiom' in the Objectivist sense) from which we can prove the various theorems that are used in mathematics for the physical sciences. And suppose we wish for the mechanics and fine points of this axiom system to be as easy to use and comprehend as possible. Well, many people feel that modern axiomatic set theory is such a system. However, suppose these axioms provide MORE than needed for the physical sciences, as the axioms provide for such things as higher and higher infinite cardinalities without end. And suppose there does not seem to be a practical way to cut the axioms down so that they give us only scientific mathematics but not higher and higher infinite cardinals. Well, that is pretty much the situation we're in. ZFC is pretty much standard modern set theory; however the somewhat weaker (Z\"regularity")+"dependent choice" (which I'll call "ZS") is considered adequate for the physical sciences. Now ZS is as described in the above paragraph. It gives us mathematics for the sciences but it overshoots by giving us greater and greater infinite sets. This set theory is like a strange high jumper who can always clear the bar at 7 feet but he can only do it by clearing 8 feet (he can't help himself from jumping at least 8 feet if he's required to jump at least 7 feet). Now, one can propose a system that does not overshoot. But how to do it and keep the complexity of working in the system manageable? Of course, 'manageable' is subjective, so it's up to each person to decide for himself whether any of the proposed alternatives are manageable.

Maybe some authors use the term 'higher order alephs', but I would just say 'higher alephs' or 'uncountable alephs'. aleph_0 is the least aleph and it is the only countable aleph. For any ordinal k>0, we have that aleph_k is a "higher aleph" or "uncountable aleph".

The definition of 'is infinite' is as I stated it. In sense (1) I mentioned, it's simply the negation of finitude: S is infinite if and only if S is not finite. It's not given through a notion of "getting there". However, there is a set theoretic notion that you might capture what you have in mind by "getting there", but that notion does not itself play a role in defining 'is infinite'. If I understand what you're getting at then yes, that is the idea. We have a definition of 'finite' and for any given S, either S meets the criteria of that definition or it does not (same for 'infinite'). If I understand you correctly, you're saying this [where 'N' stands for the set of natural numbers]: {x | x in N} is countably infinite; and {y | there is a k in N such that y=2k} is countably infinite. I.e. both the set of natual numbers and the set of even numbers are countably infinite. And that is a correct statement. I don't know, and I'm not committed to an Objectivist notion of reality such as I surmise you have in mind. I don't know what "out there" means. Whether Objectivist or not, I think we can agree that these mathematical sets are abstractions and not material objects. In what sense one wishes to say that such abstractions exist or even make sense is up to one's philosophy or framework. Objectivism rejects that there exists any "complete infinity" but accepts a notion of "potential infinity". Whether that entails that Objectivism rejects even the existence of infinite sets in the mathematical sense of "exist" as in such mathematical statements as "there exist sets that are infinite, i.e. there are sets that are examples of "complete infinities"" (commonly accepted by mathematicians) is up to Objectivists to say. However, if one withholds adopting the power set axiom, then it would not be inconsistent to hold that there exist countably infinite sets but that there do not exist uncountable sets. But, again, whether Objectivism accepts that such things as the power set axiom are even meaningful is up to Objectivists to say. Meanwhile, ordinary set theoretical mathematics does not use a notion of "potential infinity" and it's not apparent even how that notion could be formally incorporated into ordinary set theoretical mathematics.

Yes. Any set. Of course, the empty set in particular is finite. A 1-1 correspondence between S and T is a bijection from S onto T. It is not precluded that T may be a proper subset of S. S and T are in 1-1 correspondence if and only if there is 1-1 correspondence between S and T. That there are sets in 1-1 correspondence with proper subsets of themselves is easy to show by such as exmaples as this: Let 'w' stand for the set of natural numbers (i.e., the set whose members are 0, 1, 2, ...). Let f be the function whose domain is w and such that for all n in w, we have f(n) = n+1. So f is a 1-1 correspondence between w and w\{0}, i.e., between the set of natural numbers and the set of positive natural numbers. Another example, let f be the function whose domain is w and such that for all n in w, we have f(n) = 2*n. Then f is a 1-1 correspondence between the set of natural numbers and the set of even numbers. Examples abound ...

Let's set aside for the moment such things as "points of infinity" in the extended real number system. Instead, let's look at the notion of infinity as mentioned in this thread, specifically the infinitude of the set of natural numbers. In this sense, it is more accurate to refer to the adjective 'is infinite' than to 'infinity'. That is, in set theoretic mathematics, we have the property 'is infinite'. There are two prominent defintions, of which one may choose to use either: (1) S is infinite if and only if no natural number is in 1-1 correspondence with S. (2) S is infinite if and only if S is in 1-1 correspondence with some proper subset of S. In Z set theory, we have that the definiens of (2) implies the definiens of (1). That is, Z proves that if S is in 1-1 correspondence with a proper subset of itself then S is not in 1-1 correspondence with any natural number. (This is the famous "pigeonhole principle".) But not even ZF set theory proves that the definiens of (1) implies the definiens of (2). That is, ZF does not prove that if S is not in 1-1 correspondence with any natural number then S is in 1-1 correspondence with some proper subset of S. To prove that the definiens of (1) implies the definiens of (2), ordinarily some form of a choice principle is used, such as the axiom of choice or the weaker axiom of countable choice. Terminologically, "S is in 1-1 correspondence with a proper subset of S" is said as, "S is Dedekind infinite" while "S is not in 1-1 correspondence with any natural number" is said as "S is infinite" or sometimes "S is Tarski infinite".

We should first settle on a definition of 'naive set theory'. Usually naive set theory is understood to have a principle of unrestricted comprehension, so (1) the axiom of infinity is not needed as it is just an instance of unrestricted comprehension, and (2) naive set theory is inconsistent so that perforce no other axioms (such as infinity) are needed. So perhaps it is not naive set theory you have in mind but rather such theories as Z, ZF, ZFC, NBG, etc., which do have the axiom of infinity. As to alternatives, there are many in the literature, including NF, intuitionistic set theories, constructive set theories, even ultra-finitistic set theories (e.g. Levine's 'Understanding The Infinite'), and even more exotic options. However, I cannot say whether any of these meet your criteria of reduction. In any case, it seems fairly clear that there is no formal, explicitly Objectivist-approved axiomatization of the mathematics for the sciences.