An uncountable countable set
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From: MoeBlee on 30 Oct 2006 19:52 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > The point is, there are different types of numbers, and statements > > that are true of one type of number need not be true of other > > types of numbers. > Well, then, you must be of the opinion that set theory is NOT the > foundation for all mathematics, but only some particular system of > numbers and ideas: a theory. That's good. Whether he thinks set theory is or is not a foundation, it doesn't follow that he should not think it is a foundation simply because there are different kinds of numbers. MoeBlee
From: MoeBlee on 30 Oct 2006 20:02 Tony Orlow wrote: > I am beginning to realize just how much trouble the axiom of > extensionality is causing here. Oh, now the axiom of extensionality. When you buy into Robinson's non-standard analysis you buy into the axiom of extensionality, and all the other axioms of set theory, and mathematical logic - the whole kit and kaboodle - including the axiom of choice, ordinals, and uncountable cardinals, and all the "transfinitology" (even if not with platonistic committments) you so strenuously disclaim. MoeBlee
From: MoeBlee on 30 Oct 2006 20:05 Lester Zick wrote: > On 27 Oct 2006 11:43:47 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> >By having read a proof. > >> > >> A proof that there can be "no consistent theory . . .". Truly > >> fascinating. Do tell us more about this proof. > > > >I said no such thing as that there is a proof that there is no > >consistent theory. > > Especially considering your judicious pruning of the comments > involved. There's nothing I left out that justifies representing me as having claimed that there is a proof that there is no consistent theory. MoeBlee
From: MoeBlee on 30 Oct 2006 20:11 Lester Zick wrote: > On 27 Oct 2006 14:30:27 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> According to MoeBlee's recent lectures on the subject of exhaustive > >> mathematical definitions one cannot simply define IN and OUT, one must > >> use a placeholder such as IN(x) and OUT(x) to establish the domain of > >> discourse. > > > >Please stop mangling what I've said and then representing your mangled > >versions as if they are what I said. > > What's mangled about what I said? Did you or did you not say > mathematical definitions require "domain of discourse" placeholder > variables No, I did not. I explained to you that variables may be used in certain mathematical definitions to range over the domain of discourse of a given model of the theory. I didn't say that the variables "establish" the domain of discourse (which is nonsense), nor did I say that we cannot make definitions without variables (some defintions may not require variables), nor did I say that variables should be appended the way you append them and the way you've been misrepresenting the particular definition I gave. Please stop it already. MoeBlee
From: MoeBlee on 30 Oct 2006 20:28
Tony Orlow wrote: > > Do you agree that there is no logical contradiction between the fact > > that there is a least infinite ordinal and the fact that there is no > > least non-standard real number? > > I think that the concept of a least infinite number in any real sense > violates the fact that one can always remove 1 from it and it will still > be infinite. Robinson directly uses this idea. Limit ordinals directly > violate it. Is it true or not? Does removal of a nonzero quantity always > result in a smaller value, or not? That's the issue. > > So, yes, there is no deductive contradiction between the two, because > they have a difference of axiomatic assumptions to begin with. No, they don't. Robinson works in set theory and mathematical logic. And an axiomatization, such as Nelson's is a conservative extension of set theory. You are COMPLETELY mixed up. The reason infinite elements in a nonstandard model don't contradict a least infinite ordinal is that, as I've told you about a dozen times already, the word 'infinite' is being used in a different sense in the two different contexts. This is not a matter of "different axiomatic assumptions to begin with". > That > difference causes a contradiction BETWEEN the two. Wrong. Robinson works in set theory and mathematical logic. Nonstandard analysis is a result OF set theory and mathematical logic. Nonstandard analysis is not a contradiction with set theory and mathematical logic. And IST (Nelson's axiomatization of non-standard analysis) is a CONSERVATIVE extension of set theory. So any theorem of set theory is also a theorem of IST. And Robinson also talks about such things as the transfer principle. What you did was to find the words 'there is no smallest infinite number" and, without having a ghost of clue as to the actual mathematics involved and the very specific set theoretical and mathematical logical context, you seized on that as some kind of vindication of your criticisms of set theory, when actually nonstandard analysis is, quite the contrary, more credit onto the fruitfulness of set theory and mathematical logic. And you are clueless as to this, even though Robinsion says himself IN THE VERY FIRST WORDS of the book: "[...] the concepts and methods of contemporary Mathematical Logic are capable of providing a suitable framework for the development of the Differential and Integral Calculus by means of infinitely small and infinitely large numbers." And lately you've been spouting incorrectly that intuitionistic logic rejects the principle that false implies anything. Of course that is incorrect and what is the point of your even taking refuge in intuitionistic logic (of which you know nothing) if you're endorsing Robinson who uses classical logic and who avowedly rejects the restrictions of intuitionistic logic for meta-theoretic reasoning. MoeBlee |