Re: infinitely many nn's = infinite nn's?

G. Frege wrote:

On Thu, 08 Mar 2007 00:34:02 GMT, Phil <toob-headman@xxxxxxxxxxxxx> wrote:


In other words, are there any natural numbers in these unbounded sets that do NOT exist in any of the sets from 1 thru n?


Since for any n e IN the number n is in the "set from 1 thru n", for any n
e IN there is a "set from 1 thru n" such that n is in it. BUT that does not
mean that there is "set from 1 thru n" such that any n e IN is in it.

Proof: For each and any "set from 1 thru n" n+1 is not in the set. Hence
for any "set from 1 thru n" there is a number which is not in it. On the
other hand there is no (natural) number which is not in IN (by definition).

Ridiculous, look at your "reasoning," if I may be forgiven for so abusing the word. You used an argumentative structure to show that for any set 1 thru n, there is an element n+1 that is not in the set. Fine, duh, but you MUST then use that same argumentative structure to see if you can "prove" just the reverse, as in, for any element n, there is a set 1 thru n+1 that contains an element larger than n. Equally fine, equally duh, and I just "proved" that there are sets in IN which contain elements larger than any n, which is no less stupid than the conclusion from your proof, but which does have the advantage of being more obviously stupid. And just to make sure that you do not attempt to continue to be stupid with your comment that "On the other hand there is no (natural) number" -- for example, n -- "which is not in IN (by definition)," I can just as easily say that "for every (natural) number -- as in "n" -- in the set of IN, there exists a subset which contains 1 thru that number -- as in "n" -- by the definition of sets."

That's the problem with modern mathematicians, and scientists in general, they confuse argument with proof. Do you know WHY you must use the same process of reasoning, the same argumentative structure (surely there's a better term) in reverse, or in an alternate manner? A proof, as opposed to a mere argument, is something that NECESSARILY FOLLOWS from a set of premises. That means, among other things, that there are no other conclusions which can also follow from the same set of premises. If you can find two or more contradictory conclusions from a set of premises, then you cannot trust any of them. As a side note, the existence of several conclusions proves that either (1) the argumentative structure is flawed, which is probably the case with your argumentative structure (2) the premises are mutually contradictory, in the sense that they are too restrictive (3) the premises are not restrictive enough, meaning that they allow several possibilities, and you need another premise, the parallel postulates being the classic example.

Now, SOMETIMES you can work hard, and find additional TRUE proofs that will allow you to narrow down the results to just one conclusion, but one way or another, until you can get a conclusion which NECESSARILY FOLLOWS from your premises, you have nothing.

Phil


I don't know what you mean by "finite induction". The usual Peano induction axiom is

For every set K, if 0 is in K and for every natural number x in K, x+1 is also in K, then every natural number is in K.

Is this the axiom that you are using?

It is indeed.


Fine.


Should we add that it does not apply to the "unbounded sets" of the natural numbers?


No, we shouldn't. Since it _does_ apply to any set (K).


Would "all sets K" include all unbounded sets [of natural numbers] ...


Yes.


F.

.

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