An uncountable countable set
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From: Virgil on 6 Sep 2006 15:43 In article <44fe2b8f(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > >> N=S^L is inductively provable > > From what? One needs the entire arithmetic of the natural numbers before > > it even make sense, and then only makes sense for N,S and L all being > > natural numbers. > > Yes, if you include the infinite naturals. What "infinite naturals"? There are no such creatures crawling around any standard universe, so that if TO wants them, he needs must create a universe for them to exist in, which he has not yet done. > > > > So that neither "N=S^L" of "IFR" can be an axiom independent of all > > those many other axioms necessary for either to have any meaning at all. > > if they are to hold at all in any system, they must be theorems, not > > axioms. > > > > Unless they are derived from other logical statements, they're not > theorems. If the justification for the axiom is not other axioms, then > it's an axiom. But neither of those can be even stated in the absence of an axiomatic foundation, so they cannot themselves be independent of the axioms necessary to support them. > > > > > > >> The axioms of internal and external infinity are > >> constructive axioms like Peano's, and therefore somewhat "artificial" > > > > While I am not sure I recall them, I suspect that, like TO's "N=S^L" > > of "IFR", those "axioms" must be theorems, not axioms in any system in > > which they are to hold true. > > > No, they were my extension to Peano's. External: y -> x<y<z. Internal: > x<z -> x<y<z. You can insert the existential quantifiers if it makes you > more comfortable. AS they are both false for naturals, they are of no use to anyone, at least in any standard system.
From: mueckenh on 6 Sep 2006 16:25 Dik T. Winter schrieb: > > Your "each" means in symbols of logic: "A = (for) all". > > The number is nothing than all of its digit positions. > > Therefore your statement is a self contradiction. > > Where? In logical terms (A meaning "for all" and E meaning "there is"): > (1) A{p = digit position} E{q = list item} {such that q indexes p} That is your definition. But what we can safely say is only: (1') A{p = digit position of list item} E{q = list item} {such that q indexes p} > (2) A{p = digit position} E{q = list item} {such that q covers p} If your definition could be satisfied, the construction of the list would imply this, yes. What we can safely say, however, is only: (2') A{p = digit position of list item} E{q = list item} {such that q covers p} Now we may ask: Is it possible that a list item indexes or covers other digits than those which are indexed or covered by list items? The answer is: no. And we may further ask: How can 0.111... be distinguished from all list items? The answer is the only one possible: By digit positions occupied by 1's which are neither indexed nor covered by list items. Why? because 0.111... does not have any other component. So we may finally ask: Is it possible that all digit positions of 0.111... are indexed or covered by list items? The answer is obvious: no. Regards, WM
From: mueckenh on 6 Sep 2006 16:28 Virgil schrieb: > In article <1157471262.974912.294510(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1157367209.318653.75760(a)p79g2000cwp.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > Counting is the most primitive version of addition. > > > > > > Counting is not addition at all. > > > > > So you cannot even count, in your infinite unphysical mind? > > Counting precedes adding, in that one may count when unable to add. Counting is addition of 1 to n. If one is unable to add 1, one cannot count. > But how is one able to add without being able to count? That is no meaningful question in this framework. I said: Counting is the most primitive version of addition. > A primitive version of counting, from which the word "calculus" > allegedly derives, is Greek shepherds in classical times counting their > flocks by pairing off their sheep with pebbles (calculi). No. It comes from the stones of the Abacus. Newton's mathematics already had been called calculus. The history you allude to has been discovered much later (less than 100 years ago). (So you would need set theory to support your claim.) Regards, WM
From: Virgil on 6 Sep 2006 18:28 In article <1157574332.235885.113030(a)d34g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > Your "each" means in symbols of logic: "A = (for) all". > > > The number is nothing than all of its digit positions. > > > Therefore your statement is a self contradiction. > > > > Where? In logical terms (A meaning "for all" and E meaning "there is"): > > (1) A{p = digit position} E{q = list item} {such that q indexes p} > > That is your definition. But what we can safely say is only: > (1') A{p = digit position of list item} E{q = list item} {such that > q indexes p} But (1') is impossible, as one cannot specify A{p = digit position of list item} prior to specifying E{q = list item} If one wishes do Ap Eq, one cannot a priori restrict the p's to only those possible for a give q.
From: Virgil on 6 Sep 2006 18:37
In article <1157574529.452614.131100(a)d34g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1157471262.974912.294510(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <1157367209.318653.75760(a)p79g2000cwp.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > > > > Counting is the most primitive version of addition. > > > > > > > > Counting is not addition at all. > > > > > > > So you cannot even count, in your infinite unphysical mind? > > > > Counting precedes adding, in that one may count when unable to add. > > Counting is addition of 1 to n. If one is unable to add 1, one cannot > count. Counting can take place without any numbers at all, much less any arithmetic. > > > But how is one able to add without being able to count? > > That is no meaningful question in this framework. It is in my framework, as counting can be done with pebbles and no numbers a all. > I said: Counting is the most primitive version of addition. And I said, as usual, "you are wrong, as usual". > > > A primitive version of counting, from which the word "calculus" > > allegedly derives, is Greek shepherds in classical times counting their > > flocks by pairing off their sheep with pebbles (calculi). > > No. It comes from the stones of the Abacus. Newton's mathematics > already had been called calculus. The use of pebbles for counting sheep, by pairing off sheep with pebbles, precedes Newton by a couple of millennia. Similarly tally marks cut into sticks, and similar non-numerical methods of counting, can be traced back to prehistoric times. |