Schmarksvillian

'Keating' always made me think of the word 'cheating'. (I make no claims about the actual origin of 'Keating' nor about why the author of 'The Fountainhead' chose the name for the character.)

I take it you are talking about mathematics here as standard set theory. In that case, you need to define 'definite multiplicity' in terms of the primitives or previous definitions in set theory. The notion of a 1-1 correspondence does not require any notion of 'definite multiplicity'. The notion of 1-1 correspondence has a mathematical definition: S is in 1-1 correspondence with T if and only if there exists a function f whose domain is S, whose range is T, and such that for all x and y in S, if f(x) = f(y) then x=y. If by 'definite multiplicity' you mean that the cardinality operation applies to all sets, then this is proven in ZFC. Every set is in 1-1 correspondence with some ordinal, and we take the cardinality of a set to the least ordinal the set is in 1-1 correspondence. Note to moderator and others: Again I stress that various philosophical objections may be made to the notion of infinite sets. However, when a critique of the notion is given as to the sense of infinite sets in standard set theory, then that sense should not be misstated nor confused with notions that it does not include.

Cantor understood that there are "inconsistent multiplicities" (I think that is close to the term he used). But you're comment about sets being equal to a proper subset of themselves is nonsense. By definition, no set is equal to a proper subset of itself. What is the case is that some sets are equinumerous (that is, in 1-1 correspondence with) with proper subsets of themselves (as in Galileo's observation that the natural numbers are in 1-1 correspondence with the squares of natural numbers. And that is not in itself a contradiction and Cantor did not take it as such. The set of natural numbers is equinumerous with a proper subset of itself (indeed, there are many proper subsets of the set of natural numbers that are equinumerous with the set of natural numbers), as we see from Galileo's observation alone, but that is not contradictory. What Cantor referred to as "inconsistent multiplicities" were things such as the set of all sets. But Cantor's work was not axiomatic, and it is in axiomatic set theory that we find that such inconsistent multiplicities are not allowed. One is welcome to make whatever argument against the existence of infinite sets, and there are some mathematicians who reject that there exist infinite sets or who view such existence claims as nonsense. However, what you just wrote is not pertinent to the ordinary treatment of infinite sets in mathematics. In such mathematics, 'infinite' is as an ADJECTIVE. It's not that there is some set that mathematicians call 'infinity' (put aside for a moment locutions such as 'as x goes to infinity', which is another matter and in which 'infinity' drops out in full explication) but rather that certain sets that have the PROPERTY of being infinite. Then, in this mathematics (i.e., standard axiomatic set theory), it turns out that there are there are sets that are infinite but not equinumerous with one another and further that one of the sets has an injection with (a 1-1 correspondence with a subset of) the other, so we say that the second set has greater infinite cardinality than the first set. Again, I stress that this is aside from whatever other philosophical arguments one might have (whether good ones or not) against the existence or even sense of infinite sets. Merely, that your particular account of the matter is not pertinent as I've shown; you've simply not stated anything that goes on in the ordinary notions in mathematics in which the existence of infinite sets are accepted.

She allows allows concepts of method, and she allows mathematics such as complex numbers if they are useful. I don't know exactly what you mean by "infinity as some semblance with identity", but in advanced mathematics there are approaches that do reject that there exist infinite sets. There are various ultra-finitistic approaches in advanced mathematics. It's true that most mathematicians accept the existence of infinite sets, but there are some, even if only relatively a few, who work in advanced mathematics without accepting that notion, or outright rejecting it, or even claiming that it is meaningless.

Birkhoff & Mac Lane (or is it Mac Lane & Birkhoff?) is excellent, authoritative, and a classic.

No, that is NOT what I wrote in my private correction. (I don't know whether I'm allowed by the moderator to write the actual correction, but at least, for the record, I don't want the above formulation attributed to me).

It's not ill-formed. Anyway, take it in general form: Would you mention an authoritative Objectivist text that addresses the subject of 1 and .999... or anythihng closely related to it? No it doesn't. One doesn't have to affirm or deny the notions in your post just to do ordinary mathematics. Whether or not you consider the mathematical theorem: 1 = .999... to violate some notion of identity is up to you. In the meantime, the equation is well understood to say that 1 and .999 are the same number (also that '1' and '.999...' name the same number), is rigously proven in mathematics, and not even a controversial matter in ordinary mathematics.

I don't know. But I prefer ASCII since it will come out exactly as you type it.

(1) I have no comment about any of that (except I made no CONJECTURE in my post), since the plain ordinary mathematics I'm referring to does not depend on such a context. (2) Would you refer me to an authoritative Objectivist text in which it is claimed that 1 is not .999...?

Did you consult the DEFINITION of the notation '.999...'? If you don't understand that definition (I don't know whether or not you do) then little I say on the subject will likely make sense to you until you've informed yourself as to what a limit of a sequence is.

(1) You don't need quote marks around 'proof' as you use. The proof is indeed a quite ordinary mathematical proof. (2) Yes, indeed, I'm talking about mathematics and not claiming anything in the matter with Objectivism, since I've not read an Objectivist article that claims 1 is not .999... Just as, by analogy, when I mention some ordinary aspect of computing, or carpentry, or stamp collecting, I'm not opining about Objectivism. (3) Would you please mention such an Objectivist article, as I am eager to be apprised of some authoritative Objectivist decision that 1 is not the same as .999...

It's relevant as you can see the analogy from one equation to another. '1' and '2/2' name the same number, though '1' and '2/2' are different names for that number. '1' and '.999...' name the same number, though '1' and '.999...' are different names for that number. As to the thread, I've read it quite carefully. Now enough about ME and what I've read, I hope. Meanwhile, if one objects to the claim 1 = .999... on grounds that '1' and '.999...' are different names with different conceptualization involved, then does one also object to the claim 1 = 2/2 on the grounds that '1' and '2/2' are different names with different conceptualizations involved?

Who are "you guys"? Please don't claim I'm a platonist when I have not stated any platonist position.

0. No.

Just to be clear, in the proof I gave, 'infinity' is not used in a sense such as just mentioned. Whatever one's notions of mathematical infinity, the notation '-> inf' and the notions of a converging sequence and of a series are well understood in mathematics as basic as freshman calculus.

To recapitulate, it would be good to know who disagrees that the following are results of ordinary mathematics: 1 = 1 1 = 2/2 2 = (5+3)/4 1 = .999...

Would you please provide an authoritative Objectivist article in which it is objected that 1 is not the same as .999... (Of course, I don't mean in the sense that the notation '1' is different from the notation '.999...'.)

No one is disputing that the "signs" '1' and '.999...' look completely different. But 1 IS .999... no matter that '1' is a different "sign" from ".999...". '1' is a different sign from '2/2' but 1 = 2/2. '1' and '2/2' name the exact same natural number, viz. 1. That the SIGNS suggest different notions ('1' suggests the unity, or whatever, while '2/2' suggests division of the successor of unity by the successor of unity) is not in dispute; but still '1' and '2/2' NAME the same object (though, again, it is not in dispute that the NAMING is different, yet the object NAMED is the same). (By the way, the above about 1 and 2/2 modulo whatever construction of the rationals where the natural number 1 may be a rational or may only be mapped to a certain rational per a certain embedding; which does not vitiate the main point I'm making here, and I'l leave this qualification tacit for the remainder of any discussion in this context.) That's just ordinary mathematics. If you have some other philosophical qualifications, then I don't care to stop you about them, but they don't vitiate that in plain ordinary mathematics 1 IS .999...

And I have not done any such spouting. You're welcome to whatever distinction you wish to have between mathematical idenity and ontological identity. Meanwhile, in ordinary mathematics, 1 = .999..., where '=' stands for equality, which is taken in this mathematical context to mean 'the same object'. That you may have some other notion of ontological equality is not within the realm of my commentary in this particular context. I'm not talking about philosophy. I'm talking about ordinary mathematics.

That 1 = .999... doesn't depend on such notions. This is dealt with rigorously in mathematics. There are various constructions of the real numbers. In some of them, the natural number 1 turns out to be the actual real number 1, while in other constructions, the natural number 1 is mapped to (per a certain embedding function) the real number 1. The proof I showed does not depend on which of these options is chosen; the proof works in any case.

Absolutely. In my opinion, the best first approach to formal logic is to learn basic symbolic logic, or how to work in the first order predicate calculus, and for that purpose the very best book I've found is 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar. After completing that book, you can go on to more advanced subjects and I can advise you on a number of excellent texts.

To whom it may concern: Here again is a rigorous proof. If you do not understand this proof then you don't understand the basics of this subject and you'd do better to STUDY the subject rather than ignorantly SPOUT off about it: Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j). Let f(k) = SUM(j = 1 to k) 9/(10^j). Show that lim(k = 1 to inf) f(k) = 1. That is, show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e. First, by induction on k, we show that, for all k, 1 - f(k) = 1/(10^k). Base step: If k = 1, then 1 - f(k) = 1/10 = 1(10^k). Inductive hypothesis: 1 - f(k) = 1/(10^k). Show that 1 - f(k+1) = 1/(10^(k+1)). 1 - f(k+1) = 1 - (f(k) + 9/(10^(k+1)) = 1 - f(k) - 9/(10^(k+1)). By the inductive hypothesis, 1 - f(k) - 9/(10^(k+1)) = 1/(10^k) - 9/(10^(k+1)). Since 1/(10^k) - 9/(10^(k+1)) = 1/(10^(k+1)), we have 1 - f(k+1) = 1/(10^(k+1)). So by induction, for all k, 1 - f(k) = 1/(10^k). Let e > 0. Then there exists n such that, 1/(10^n) < e. For all k > n, 1/(10^k) < 1/(10^n). So, |1 - f(k)| = 1 - f(k) = 1/(10^k) < 1/(10^n) < e.

We don't need any unfinished process of division to show that 1 = .999...

1 and .999... are the exact same mathematical object. We may have different notions about the names '1' and '.999...' but they are two names of the same object.

Your parenthetical inference comes from your own rather rough simplification of Kant's theory. I am interested in what particular passages in Kant you take as advocating that one should regard oneself as nothing but a means. I'm not proposing any particular interpretation of Kant. In this instance I'm just trying to find what passages you regard as Kant advocating that one should regard oneself as nothing but a means. I don't need Google search results. I'm just inquiring as to what particular passages in Kant YOU think advocate that one should regard oneself as nothing but a means. Of course, if you wish not to state what particular passages you have in mind, then so be it.