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§ 335
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WM
Sep 7, 2013, 5:51:40 AM9/7/13
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§ 335
Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?"
(cp. § 332) has raised some comments by Andrej Bauer:
He {{J.D. Hamkins}} did not say that it is consistent to "postulate in ZFC that undefinable numbers do not exist". What he was saying was that ZFC cannot even express the notion "is definable in ZFC". {{Therefore it is very surprising that ZFC is allegedly able to define sets and numbers, i.e., to do things that nobody in ZFC can prove to be done correctly.}}
Joel made a very fine answer, please study it carefully. Joel states that there are models of ZFC such that every element of the model is definable {{although nobody can know precisely what that means.}}. This does not mean that inside the model the statement "every element is definable" is valid. The statement is valid externally, as a meta-statement about the model. Internally, inside the model, we cannot even express the statement. {{And externally we cannot find out whether external statements are meaningful, because "external" is also only some model - yet a bigger one.}}
Annix answered: I do not say undefinable numbers do not exist. Their existence follows from axiom of choice {{perhaps it follows, but as a contradiction, because you cannot choose one of many undefined numbers}} and in theory we can uniquely define each undefinable number by specifying infinite number of its properties. The problem is that the theorems of analysis as taught in universities sufficiently rely on the properties of definable numbers. {{That is not a problem of mathematics since other numbers have no properties. Also it is not a "problem" of ZFC but a simple contradiction in ZFC. Of course Zermelo would not have been stupid enough to defended in his 1908 paper the axiom of choice in length as a natural choice if he had been confronted with undefinable numbers. That nonsense has only become en vogue in the circles of modern "logicians". Of course nobody can say what real number is undefinable. Why don't the undefinable-number-cranks believe in undefinable natural numbers? Of course nobody can say what natural number is undefinable. But then countability-spook would no longer haunt those poor peolple's mind.}}
Regards, WM
Annix's question "Is the analysis as taught in universities in fact the analysis of definable numbers?"
(cp. § 332) has raised some comments by Andrej Bauer:
He {{J.D. Hamkins}} did not say that it is consistent to "postulate in ZFC that undefinable numbers do not exist". What he was saying was that ZFC cannot even express the notion "is definable in ZFC". {{Therefore it is very surprising that ZFC is allegedly able to define sets and numbers, i.e., to do things that nobody in ZFC can prove to be done correctly.}}
Joel made a very fine answer, please study it carefully. Joel states that there are models of ZFC such that every element of the model is definable {{although nobody can know precisely what that means.}}. This does not mean that inside the model the statement "every element is definable" is valid. The statement is valid externally, as a meta-statement about the model. Internally, inside the model, we cannot even express the statement. {{And externally we cannot find out whether external statements are meaningful, because "external" is also only some model - yet a bigger one.}}
Annix answered: I do not say undefinable numbers do not exist. Their existence follows from axiom of choice {{perhaps it follows, but as a contradiction, because you cannot choose one of many undefined numbers}} and in theory we can uniquely define each undefinable number by specifying infinite number of its properties. The problem is that the theorems of analysis as taught in universities sufficiently rely on the properties of definable numbers. {{That is not a problem of mathematics since other numbers have no properties. Also it is not a "problem" of ZFC but a simple contradiction in ZFC. Of course Zermelo would not have been stupid enough to defended in his 1908 paper the axiom of choice in length as a natural choice if he had been confronted with undefinable numbers. That nonsense has only become en vogue in the circles of modern "logicians". Of course nobody can say what real number is undefinable. Why don't the undefinable-number-cranks believe in undefinable natural numbers? Of course nobody can say what natural number is undefinable. But then countability-spook would no longer haunt those poor peolple's mind.}}
Regards, WM
Virgil
Sep 9, 2013, 12:51:26 PM9/9/13
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In article <8c8e5ff8-25b1-4dca...@googlegroups.com>,
What he teaches may be, but what he believes in is not.
--
WM <muec...@rz.fh-augsburg.de> wrote:
> § 335
>
> Annix's question "Is the analysis as taught in universities in fact the
> analysis of definable numbers?"
The analysis as understood by WM is certainly not that of mathematics.
> § 335
>
> Annix's question "Is the analysis as taught in universities in fact the
> analysis of definable numbers?"
What he teaches may be, but what he believes in is not.
--
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