GrandMinnow

I would like to see a direct quote of Hilbert on that. 
Hilbert did discuss that, in one way, formal systems can be viewed separately from content or meaning. But that does not imply that in another way they cannot be viewed with regard to content or meaning. Indeed, Hilbert was very much concerned with the "contentual" aspect of mathematics. 
Granted, descriptions of Hilbert as viewing mathematics as merely "a pure game of symbols", "without meaning", et. al do occur in literature that simplifies discussion of Hilbert. But for years I have asked people making the claim (here moderated to "reliability") to provide a direct quote from Hilbert. And just looking at Hilbert briefly is enough to see that he was very much concerned with the contentual in mathematics.
I'm simplifying somewhat, but Hilbert distinguished between (1) statements that can be checked by finitistic means and (2) statements that cannot be checked by finitistic means.
Finitistic means are those that can be reduced to finite counting and combination operations - even reducing to finite manipulations of "tokens" (such as stroke marks on paper if we need to concretize). This is unassailable mathematics, even for finitists and constructivists. If one denies finitistic mathematics, then what other mathematics could one possible accept?
On the other hand, mathematics also involves discussion of things such as infinite sequences (try to do even first year calculus without the notion of an infinite sequence). So Hilbert wanted to find a finitistic proof that our axiomatizations of non-finitistic mathematics are consistent. So, there would be unassailable finitistic mathematics (which has clear meaning - that of counting and finite combinatorics) and there would be axiomatized non-finitistic mathematics (of which people may disagree as to whether it has meaning and, if it does have meaning, what that meaning is) that would at least have a finitistic proof of its consistency.
So, of course Hilbert regarded finitistic mathematics as having meaning and being completely reliable. And, I'm pretty sure you will find that Hilbert also understood the scientific application of non-finitistic mathematics (such as calcululs). But he understood that it cannot be checked like finitistic mathematics; so what he wanted was a finititistic (thus utterly reliable) proof that non-finitistic mathematics is at least consistent.
However, Godel (finitistically) proved that Hilbert's hope for a finitistic consistency proof cannot be realized. 
Regarding looking at formal systems separately from content: Imagine you have a formal system such as a computer programming language. We usually regard it to have meaning, such as the actual commands it executes on physical computers or whatever. But also, we can view the mere syntax of it separately, without regard to meaning. One could ask, "Is this page of code in proper syntax? I don't need to know at this moment whether it works to do what I want it to do; I just need to know, for this moment, whether it passes the check for syntax." 
So formal symbol rules can viewed in separation from content, or they can also be viewed with regard to content. Hilbert emphasized, in certain context the separation from content, but in so doing, he did not claim that there is not also a relationship with content.

Mathematicians, and different mathematicians, mean different things depending on context. The context is either stated explicitly or reasonably gleaned per a given book or article.
So, just to narrow down, let's look at just two of the different contexts. (They are different but they support each other anyway.) To avoid getting too complicated for the purposes of brief posting, I'll give only a sketch, leaving out a lot of details, and not explain every concept (such as 'free variable') and taking some liberties with the notation and concepts, and for ease of reading, I won't always include quote marks to distinguish mention as opposed to use. (So this is not as accurate as a more authoritative treatment).
So two contexts:
(1) General, informal (or informal mixed with formal) discussion in mathematics about natural numbers.  (2) Formal first order Peano arithmetic [I'll just call it 'PA' here].
 
(1) In general mathematics, we might taken commutativity of addition to be obvious and thus a given. Or one might say:
 
"Okay, I'm going to state some truths about natural numbers from which I can prove a whole bunch of other truths, even though they're obvious anyway. The truths about addition I want to mention are:
0 added to any number is just that number. In symbols: x+0 = 0.
The sum of a number and the successor of another (or same) number is just the successor of the sum of the number and the other number. In symbols: x+Sy = S(x+y), or, put another way (where 'S' is defined as '+1'), x+(y+1) = (x+y)+1.
The induction rule. 
Now, with those three truths, one of the many truths I can prove, without assuming anything about natural numbers or what they are, other than those three truths, is the commutativity of addition. In whatever way you conceive the natural numbers, as long that conception includes those three truths I just mentioned, then the commutativity of addition is proven true."
 
Notice that we can't do this with the real numbers, because the induction rule does not work for the real numbers. So, for real numbers, we would take commutativity as an axiom (or in set theory, we would prove commutativity from the properties of the real numbers as they are set theoretically "constructed"). 
 
(2) PA, as a system, has a formal first order language, with the primitive logical symbols (including '='  as a logical symbol) and certain primitive non-logical symbols.
 
The logical symbols are:
Infinitely many variables: x, y, etc.
-> (interpreted as the material conditional)
~ (interpreted as negation)
and, from '->' and '~' we can define:
& (interpreted as conjunction)
v (interpreted as inclusive disjunction) 
A (so that, where P(x) is any formula with 'x' occurring free, AxP is always interpreted as "for all x, P(x)") 
and, from 'A' and '~' we can define:
E (so that ExP(x) is always interpreted as "there is an x such that P(x)")
 
The non-logical symbols are :
0
S
+
*
We define 
S(0) =1
S(1) = 2
etc. 
When the language is interpreted: '0' is assigned to a particular member of the domain of the interpretation; 'S' is assigned to a 1-place function (operation) on the domain, '+' and '*' are each assigned to 2-place functions on the domain. 
With the "intended" ("standard") interpretation: the domain is the set of natural numbers, '0' is assigned to the number zero, 'S' is assigned to the successor operation, and '+' and '*' are assigned to the addition and multiplication operations respectively. 
And, since '=' is a logical primitive, we assign it to the identity (equality) relation on the domain. 
So for any interpretation (such that each variable, in its role as a free variable, is assigned to some member of the domain):
x+y
is assigned to the value of the '+' operation applied to the ordered pair: <the assigned value of x, the assigned value of y>.
And 
x+y = y+x
holds in the interpretation if and only if the value of x+y is identical with (is equal to) the value of y+x.

So, to answer your question, in the syntax of the formal system itself, nothing is assumed as to what 'x' and 'y' stand for. But with a formal interpretation of the system, 'x', as a free variable stands for some member of the domain and 'y',  as a free variable, stands for some member of the domain. And with the standard interpretation, the domain is the set of natural numbers.
However, often we tacitly understand that when formulas such as x+y = y+x are asserted, we take that assertion to be the universal closure:
AxAy x+y = y+x (abbreviated Axy x+y = y+x)
And so, with the standard interpretation, that asserts that addition is commutative. And we prove it from the PA axioms (we only need the three I mentioned in a previous post, which correspond to the three truths I mentioned in this post).

Some topics that have been mentioned:
(1) MATHEMATICAL INDUCTION on NATURAL NUMBERS
Induction [by 'induction' in such contexts, I mean mathematical induction] is ordinarily used in these contexts:
* Proofs from the axioms of Peano arithmetic [by 'Peano arithmetic' I mean first order Peano arithmetic] in which induction is an axiom. Induction is needed because there are many things you can't prove about natural numbers from the Peano axioms without the induction axiom.
* Proofs from the axioms of set theory in which there is the set of all and only the natural numbers and that set admits induction. Induction is used  because it is the induction property of the natural numbers that permits many of the proofs about natural numbers.
* Proofs historically before Peano arithmetic or set theory. (But such proofs can be put in Peano arithmetic or set theory retroactively.) 
* Proofs in general mathematics in instances where Peano arithmetic or set theory are not necessarily explicitly mentioned. (But said mathematics can be formulated in Peano arithmetic or set theory.)
And none of this stems from any supposed need to avoid "derailment" from infinite cardinals or ordinal addition.  

(2) USE/MENTION
There is a distinction between a) symbols, or sequences of symbols that are terms, to stand for objects or range as a variables over objects and b) the objects that are symbolized. 
Single quote marks indicate that a linguistic object - a symbol or sequence of symbols - is referred to. (Actually, more exactly, for sequences we would use a concatenation marker, but that is too pedantic for this discussion.)
'2' is a symbol (a linguistic object), it is not a number. However, 2 is a number. 
But this has really nothing to do with stating the commutativity of addition. 

(3) IDENTITY
x = y 
means 
x and y are the same object. 
So '=' stands for the identity (equality) relation. 
If T and S are terms, then 
T = S 
means that T and S both name the same object.
Equivalence was mentioned. The identity relation is an equivalence relation, but there are equivalence relations other than identity. But there is nothing gained in this discussion by mentioning a warning against confusion with equivalence relations. There is no mistaking that '=' stands for identity.  
 
(4) An article titled 'Infinity plus one' was linked to. The title of that article is misleading. In regards to cardinals, we don't use 'infinity' as a noun, but rather 'is infinite' as an adjective. (This is different from such things as "points of infinity" in the extended reals system, as such points don't refer to cardinality but rather to ordering.)
 
(5) This comment was posted: "It was posited that the equation (1+a=a+1) could not be verified, because we would need to check it against every possible number, which is impossible to do because infinity." Just to be clear, that is not necessarily my own view, but rather it was part of a brief explanation of Hilbert's views, and even in that regard, the statement needs important qualifications such as those I mentioned. 

I have to emphasize that I am not a scholar on Hilbert, mathematics, or philosophy, so my explanations are not necessarily always perfectly on target, and at a certain depth, I would have to defer to people who have studied more extensively than I have. And I don't mean necessarily to defend Hilbert's philosophical notions in all its aspects.
That said, however, here's a stab at answering your question:
I think what Hilbert has in mind is the distinction between a) reasoning with symbols that are taken as representing particular numbers and b) making generalizations about an infinite class of numbers.
For example, if 'a' is a token for a particular number, then the truth of 'a+1 = 1+a' cannot be reasonably contested as it can be concretely verified - it is finitistic. For example, for the particular numeral '2', the truth of '2+1 = 1+2' cannot be reasonably contested as it can be concretely verified.  
On the other hand, where 'A' stands for any undetermined member of entire infinite class of numbers, then 'A+1 = 1+A' (which is ordinarily understood as 'for all numbers A, we have A+1 = 1+A') cannot be verified concretely because it speaks of an entire infinite class that we can't exhaustively check. Therefore, some other regard must be given the formula. And that regard is to take it as not "contentual" but as "ideal" but formally provable from formal axioms (which are themselves "ideal"). And it is needed that there is an algorithm that can check for any purported formal proof that it actually is a formal proof (i.e., that its syntax is correct and that every formula does syntactically "lock" in sequence in applications of the formal rules); this is what Hilbert has in mind as the formal "game". Then Hilbert hoped that there would be found a formal proof, by using only finitistic means, that the "ideal" axioms sufficient for ordinary mathematics are consistent. Godel, though, proved that Hilbert's hope cannot be realized. 

Some of these points have been mentioned, but I'd like to summarize:
 
(1) P -> ~P, as ordinarily understood, is a formula of symbolic logic, which is a context that may differ from the Objectivist notion of logic. One cannot understand such formulas of symbolic logic without first studying the textbook basics of symbolic logic (the Kalish-Montague-Mar book that was mentioned is indeed a fine introductory text). It is not meaningful to discuss such formulas with only a "kinda sorta" vague understanding mixed with Objectivist terminology that might not apply to the specialized terminology of symbolic logic.
 
(2) Symbolic logic has different systems. The usual context of such a formula is what is called 'classical sentential logic' (or 'classical propostional logic'). A less usual context is intuitionistic sentential logic. And there are others. But for purposes of basic discussion, I'll keep to the context of classical sentential logic.
 
(3) Reidy is incorrect that (P -> ~P) -> (P & ~P) is a theorem. What is, for example a theorem, is (P -> ~P) -> ~P. (Also, his mention of the completeness theorem in this context is wrong.)
 
(4) The letter 'P' here is a variable that ranges over "statements" (more precisely 'formulas'). It does not range over other objects. So conflating '->' with 'is' makes no sense.
 
(5) The symbol '->' stands for the Boolean function that maps <P Q> to 0 when P is mapped to 1 and Q is mapped to 0, and maps <P Q> to 1 otherwise. (And '0' can be interpreted as 'false' and '1' as 'true'.) 
 
In this sense, 'P -> Q' is understood as 'if P then Q' (this is called the 'material conditional'). P -> Q is false when P is true and Q is false, and it is true otherwise. 
 
This 'if then' is not claimed to correspond to all other English language meanings of 'if then'. It does correspond usually, but not always, and it is not meant to always correspond. In particular, some people find it wrong, or at least odd, that P -> Q is true when P is false. But in context, it is not intended that this sense of 'if then' corresponds always with certain other ordinary English senses, though it does correspond in a basic way, in the sense of the following analysis:
 
We do not need '->' in symbolic logic. We could take it as a defined symbol in this way:
 
P -> Q
by stipulative definition of our specialized symbol '->' is merely an abbreviation for 
~(P & ~Q).
 
And it is easy to see that, even in virtually all everyday English contexts, ~(P & ~Q) is false when P is true and Q is false, and it is true otherwise, just as we said for P -> Q. 

No, P -> ~P does not just mean "some true proposition entails another true proposition that's not the same as the first proposition."
 
Rather, it means, "If P is true then P is false".
 
And it is not a contradiction. Rather, it boils down to saying "P is false". It does NOT say that there is a statement P such that P is both true and false. '->' is NOT the same as '&'. It says that P implies its own negation, so P itself implies a contradiction since P implies itself (of course) but also it implies its own negation. So P is false since P implies a contradiction. 
 
Again, P -> ~P is not a contradiction. What is a contradiction is to assert both P and P -> ~P.
 
Please, I wish all the people in this thread who are opining about this subject without FIRST learning the basics from a textbook, would indeed first read the chapters of a texbook and then come back to discuss it.

(1) Those passages don't quote Hilbert or cite any reference to his texts.
(2) The passages are from what I think might be a popularizing book [Godel: A Life In Logic] on the subject. Often such popularizations misleadingly oversimplify the subject. Without having read the book, I won't claim that it does misleadingly oversimplify, but I would caution to look out for possible oversimplifications. That set of passages onto itself might be okay yet it could stand some explanation.
(3) Anyway the passages don't say or even imply that Hilbert took mathematics as entirely a meaningless game of symbols. 
(4) And not only do those passages not say or imply that Hilbert took mathematics as entirely a meaningless game of symbols, but the passages say the OPPOSITE.
/
I am not an authority on this subject; I have read only some of Hilbert's translated writings and none of his writings in German that remain untranslated to English. So my own comments may be too simple or require qualification or sharpening. For a first reference on the Internet, I would suggest:
http://plato.stanford.edu/entries/hilbert-program/
http://plato.stanford.edu/entries/formalism-mathematics/
Moreover, a few years ago, one of the contributors to the Foundations Of Mathematics Forum asked whether anyone knows of any attribution to the writings of Hilbert in which he said that mathematics is only a game of symbols. As I recall, at that time, no one did. (Posters on The Foundations Of Mathematics Forum are almost entirely scholars in the field of mathematical logic and the philosophy of mathematics.)
That said, here are some general points:
(1) Hilbert recognized the role of mathematics in the sciences. He would not regard mathematics as merely a symbol game. 
(2) Hilbert may regard formal systems as subject to being taken, in certain respects, as without meaning. However, I know of no attribution in which Hilbert claimed that mathematics is merely formal systems. Moreover, Hilbert recognized that, while in one aspect formal systems are to be regarded as without meaning, in other aspects, formal systems are to receive interpretation and in interpretation we evaluate meaning. 
The rough idea is that syntax onto itself is without meaning but with semantics we do evaluate meaning.
The syntax includes the formation rules for formulas and the rules for proof steps. Syntax is regarded onto itself as without meaning so that no "subjective", vague, or inexact considerations are allowed in checking whether a symbol string does obey the formation rules for formulas or whether a purported formal proof does indeed use only allowed inference rules. For example, regarding formation rules, when you run a syntax check on lines of computer program code, the syntax checker doesn't care about the "meaning" of your code (say, for example, what it will accomplish for the user of the application or whether the user will like the results, etc.) but only whether the code follows the exact rules of the syntax of the programming language. 
The semantics include the interpretation of the symbols and of the formulas made from the symbols. This is meaning. The interpretation itself can be done either in formal or informal mathematics. For example:
Ax x+0 = x
This is formal string of symbols that in itself has no meaning.
But with a semantics that specifies the domain of natural numbers and interprets 'A' as 'all', 'x' as a "pronoun", '+' as the operation of addition, '0' as the natural number zero, and '=' as identity, we have the interpretation:
zero added to any number is that number
Of course, that example is so simple as to make the method seem silly; with more complicated formulations we see the advantage of the method.
(3) Also, Hilbert distinguished between the contentual and the ideal in mathematics.
Most basicially, the contentual is the the finitary mathematics of "algorithmic" operations on natural numbers. This was later articulated as the formal system PRA (primitive recursive arithmetic), though Hilbert's own earlier work was in a different but akin system. Such operations on natural numbers can be mutually understood as operations on finite strings of symbols. 
The ideal are the infinitary notions of set theory that is used to axiomatize real (number) analysis, as with analysis we regard infinite sequences, etc. 
Hilbertian formalism ("Hilbert's program") is:
The finitary is "safe" and unimpeachable. But, while the ideal may itself be without contentual meaning, it is used as a formal framework for deriving formal theorems (that are later interpreted as generalizations regarding natural numbers and also for real analysis). Then, we wish to know whether the finitary mathematics can prove that the infinitary mathematics is consistent (without formal self-contradiction).
It is Hilbert's hope and expectation of such a finitary proof of the consistentency that was proven by Godel to be unattainable. With regard to Hilbert's program, Godel's second incompleteness theorem reveals that there is no finitary proof of the consistency of the theory of natural numbers (generalizing beyond Godel's particular object theory, say, for example, first order Peano arithmetic) let alone of real analysis.
/
Now let's look at some of the passages from that book:
(1)  "getting at the mathematical truth "
Truth pertains to meaning. If Hilbert was concerned with "getting at the mathematical truth", then he could not have regarded mathematics as merely meaningless symbol manipulating.
(2)  "the statements (symbol strings) should be paradox-free.  In particular, there should be no undecidable propositions "
I don't know all that's intended there, but (un)decidability is a separate (though in certain ways, related) question from consistency (consistency being a formal counterpart to "paradox-free"). 
(3)  "how to interpret the meaningful mathematical objects in terms of meaningful formal ones"
I might say that what are meaningful or not are not objects but instead formulas (or even notions). In any case, again, we see that Hilbert is indeed concerned with meaning. Notions about ideal objects may not be meaningful, but notions about contentual objects are meaningful. PRA has an immediate and "concrete" meaningful interpretation. Then other systems give rise to abstract infinitary notions that don't have such concrete meaning but are "residue" of said formal system that provides theorems regarding generalizations with finitary mathematics and real analysis (which is the theory of the real number calculus used as the basic mathematics of the sciences). Hilbert hoped further that finitary mathematics would prove the consistency of infinitary mathematics - but that's the part proven by Godel not to be possible.
(4)  "Hilbert didn't believe that any Russell-type paradoxes [Set Paradox, Barber Paradox, etc.] lurked in the world of mathematical truths, even though they might exist in the far fuzzier realm of natural language"
Hilbert would have easily known that the Russell paradox can occur even in a formal system (most saliently, Frege's system). Formalization itself does not ensure consistency. 
(5)  "And the way he thought we could prevent them from crossing the border separating ordinary language from mathematics was to formalize the entire universe of mathematical truth.  What Godel showed was that Hilbert was dead wrong."
Hilbert hoped for (indeed, expected) a consistent and complete formal axiomatization of the arithmetic of natural numbers and of analysis. By 'consistent' we mean that there is no formal sentence of this system such that both the sentence and its negation can be proven in the system. By 'complete' we mean that for every formal sentence of this system, either the sentence or its negation can be proven in the system, thus that the sentence is decidable, i.e. that there is an algorithm to decide whether there exists a proof in the system of the sentence (for example, we could keep running proofs until we reach one that either proves the sentence or proves its negation). Godel proved that that expectation was wrong. But this does not entail that formal axiomatizations are not still of great value and interest, as indeed the vast amount of "ordinary" analysis is formalized in any of various formal systems.

Consider sets of flowers. There are many different sets of flowers. For example, the set of flowers in my back yard, the set of flowers given to my mother over the years, the set of flowers whose only members are the largest flower in the White House Rose Garden and the flower I just saw on my co-worker's desk.
 
These are:
 
{x | x is a flower in my back yard}
 
{x | x is a flower given to my mother during her life}
 
{x | x is the largest flower in the White House Rose Garden}
 
{x | x is the largest flower in the White House Rose Garden or x is the flower on my co-worker's desk}
 
And there is the set of all flowers:
 
{x | x is a flower}
 
And with numbers:
 
{x | x is a natural number less than 5}
 
{x | x is a solution to the equation x^2 = 4} 
 
{x | x is an even number}
 
{x | x is a natural number}
 
And {x | x is a natural number} is the set of all natural numbers (or we just say 'the set of natural numbers' as it is tacitly understood that we mean 'all').
 
So "the set of natural numbers" is not the same as just saying "numbers". It's not even grammatical to take them as the same:
 
"The set of natural numbers has no greatest member" is not even expressed grammatically by "Numbers has no greatest member".
 
/
 
As to infinity, one will not get a coherent discussion by mixing up two separate contexts: Objectivism and standard mathematics. I don't opine about Objectivism, but I should say what 'infinite' refers to in standard mathematics:
 
First, there is no object called 'infinity' (putting aside the "point of infinity" on the extended real line, which is something else). Instead there is 'infinite', which is an adjective not a noun. So it makes sense to say "the set of natural numbers is infinite" since the adjective 'is infinite' is true of the set of natural numbers. But we don't say "infinity exists" or "infinity is a number" or any of that.
 
Now, for any set S (such as the set of natural numbers) "S is infinite" means "S is not finite", which in turn is equivalent to "there is no 1-1 function between S and a natural number" (another definition that is equivalent with the axiom of choice is "there is no 1-1 function between S and a proper subset of S)".
 
Meanwhile, while there is no object called 'infinity', we sometimes loosely say "the first infinity" to mean "the first infinite cardinal number" (which is itself a set) and things like that. But that is a loose way of speaking and we don't actually mean that there an object called 'infinity', just that there is an object that is the first infinite cardinal and other infinite cardinals and infinite sets in general. 
 
Put another way, a set may have the PROPERTY of being infinite. But there is no THING that we call 'infinity'. 
 
/
 
Granted, there are mathematicians (called 'strict finitists') who do not allow systems in which there are infinite sets. But in the usual instances, mathematicians regard infinitude along the lines I just described and they do accept that there are infinite sets, or at the very least that the axioms of set theory prove the formula that is rendered into English as "there exist infinite sets". 

Some of these points have been mentioned, but I'd like to summarize:
 
(1) P -> ~P, as ordinarily understood, is a formula of symbolic logic, which is a context that may differ from the Objectivist notion of logic. One cannot understand such formulas of symbolic logic without first studying the textbook basics of symbolic logic (the Kalish-Montague-Mar book that was mentioned is indeed a fine introductory text). It is not meaningful to discuss such formulas with only a "kinda sorta" vague understanding mixed with Objectivist terminology that might not apply to the specialized terminology of symbolic logic.
 
(2) Symbolic logic has different systems. The usual context of such a formula is what is called 'classical sentential logic' (or 'classical propostional logic'). A less usual context is intuitionistic sentential logic. And there are others. But for purposes of basic discussion, I'll keep to the context of classical sentential logic.
 
(3) Reidy is incorrect that (P -> ~P) -> (P & ~P) is a theorem. What is, for example a theorem, is (P -> ~P) -> ~P. (Also, his mention of the completeness theorem in this context is wrong.)
 
(4) The letter 'P' here is a variable that ranges over "statements" (more precisely 'formulas'). It does not range over other objects. So conflating '->' with 'is' makes no sense.
 
(5) The symbol '->' stands for the Boolean function that maps <P Q> to 0 when P is mapped to 1 and Q is mapped to 0, and maps <P Q> to 1 otherwise. (And '0' can be interpreted as 'false' and '1' as 'true'.) 
 
In this sense, 'P -> Q' is understood as 'if P then Q' (this is called the 'material conditional'). P -> Q is false when P is true and Q is false, and it is true otherwise. 
 
This 'if then' is not claimed to correspond to all other English language meanings of 'if then'. It does correspond usually, but not always, and it is not meant to always correspond. In particular, some people find it wrong, or at least odd, that P -> Q is true when P is false. But in context, it is not intended that this sense of 'if then' corresponds always with certain other ordinary English senses, though it does correspond in a basic way, in the sense of the following analysis:
 
We do not need '->' in symbolic logic. We could take it as a defined symbol in this way:
 
P -> Q
by stipulative definition of our specialized symbol '->' is merely an abbreviation for 
~(P & ~Q).
 
And it is easy to see that, even in virtually all everyday English contexts, ~(P & ~Q) is false when P is true and Q is false, and it is true otherwise, just as we said for P -> Q. 

Objecitvism accepts infinity only as potential. So I can't even imagine in what sense an Objectivist would take the continuum hypothesis as even a meaningful statement.