albrecht
Posts:
1,136
Registered:
12/13/04
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Re: Uncountability of the Real Numbers Without Decimals
Posted:
Dec 5, 2013 9:45 AM
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On Tuesday, December 3, 2013 7:28:58 PM UTC+1, Zeit Geist wrote:
> On Tuesday, December 3, 2013 4:09:52 AM UTC-7, Albrecht wrote:
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> > On Monday, December 2, 2013 9:09:39 AM UTC+1, Zeit Geist wrote:
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> > You are right with the fact that there is no sequence X = { x_n | n e N } of all reals. If you define this fact as Uncountability, the reals are uncountable. Okay.
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> > But does this mean that there are more reals than naturals? No.
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> > Without Axiom of Infinity and Axiom of Powerset: Anyway: No.
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> > And with AoI and AoP: No. Because ZFC has to have a countable model and the quantifiers "less" and "more" are irrelevant in the domain of infinity.
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> > Infinity/Ceaselessness is a quality , not a quantity.
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> A fine Philosophy, but I work in the context of Modern Mathematics. Usually within a Model of ZF(C). Here Y is "more than" X is defined as:
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> 1. There exists a Surjective Function from Y to X.
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> 2. No Function from X to Y is a Surjection.
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> Even though the Model may only be Countable, there is NO Surjection from N to R in the Model. Hence, there are "more" Reals than Naturals. A Surjection from R to N is trivial.
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> ZG
It is useless to discuss a proof if one of the opponent claims axioms which imply the disputable result. If you work in the context of ZF(C), your "proof" is nothing more than self-evident.
But this is typical: First claiming axioms of finishable infinity and its powersets - and then - heureka - there are different infinities.
Actually, this kind of reasoning is loughable.
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