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 Post subject: AxiomsPosted: Tue, 17 Jul 2012 02:34:25 UTC
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Suppose we are constructing a proof and we want some integers and to satisfy finitely many conditions at once, do we need the axiom of choice, or a special case of the axiom of choice?

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 02:48:52 UTC
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glebovg wrote:
Suppose we are constructing a proof and we want some integers and to satisfy finitely many conditions at once, do we need the axiom of choice, or a special case of the axiom of choice?

Finite choice is no problem --- just concatenate the choices in a conjunction.

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 03:37:48 UTC
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There are infinitely many integers and we only want particular integers to satisfy finitely many conditions. Are we guaranteed to find such integers without invoking any axioms?

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 03:56:34 UTC
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glebovg wrote:
There are infinitely many integers and we only want particular integers to satisfy finitely many conditions. Are we guaranteed to find such integers without invoking any axioms?

If you want to verify that two integers satisfy something, you won't need anything, just try and grab pairs and check to see if they work.

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 03:58:12 UTC
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But what if they are arbitrary and I am supposing they satisfy finitely many conditions?

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 04:07:04 UTC
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glebovg wrote:
But what if they are arbitrary and I am supposing they satisfy finitely many conditions?

In theory the integer axioms tell you what integers are, but that's REALLY PEDANTIC. If you have simple conditions you just list off integer pairs and check them. Unless they're REALLY WEIRD you can probably make a computer do it.

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 04:31:05 UTC
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Let and suppose is an arbitrary large integer. What if, for example, we want to choose s.t. all hold? I cannot use a computer because all integers are arbitrary.

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 04:37:49 UTC
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glebovg wrote:
Let and suppose is an arbitrary large integer. What if, for example, we want to choose s.t. all hold? I cannot use a computer because all integers are arbitrary.

If exists, then the set

exists by axiom scheme of separation (you can write it out in full if you want). No axiom of choice is needed.

On the other hand, to assert is nonempty, you need to dig into the arithmetics of , which again you don't need AC to see that or something like that works.

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 04:42:08 UTC
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Could you briefly explain the axiom schema of separation? I have not studied ZFC.

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 04:48:56 UTC
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glebovg wrote:
Could you briefly explain the axiom schema of separation? I have not studied ZFC.

The axiom of separation is the axiom that says (loosely) given a set and some property , there exists a set which picks out those elements of such that holds . It is called axiom scheme because in reality the axiom depends on so this is actually a collection of axioms , one for each .

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 04:51:07 UTC
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outermeasure wrote:
glebovg wrote:
Let and suppose is an arbitrary large integer. What if, for example, we want to choose s.t. all hold? I cannot use a computer because all integers are arbitrary.

If exists, then the set

exists by axiom scheme of separation (you can write it out in full if you want). No axiom of choice is needed.

On the other hand, to assert is nonempty, you need to dig into the arithmetics of , which again you don't need AC to see that or something like that works.

What do you mean? You just need to have it check based on your choice of k and N and q, just put a well-ordering on it and set it off. It may take a while, but if such exist it should find it eventually.

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 Post subject: Re: AxiomsPosted: Tue, 17 Jul 2012 17:57:45 UTC
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outermeasure wrote:
glebovg wrote:
Could you briefly explain the axiom schema of separation? I have not studied ZFC.

The axiom of separation is the axiom that says (loosely) given a set and some property , there exists a set which picks out those elements of such that holds . It is called axiom scheme because in reality the axiom depends on so this is actually a collection of axioms , one for each .

I found a separation theorem in a ZFC book, which pretty much proves the above (using some assumptions). Is that an older formulation of ZFC.

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 Post subject: Re: AxiomsPosted: Wed, 18 Jul 2012 23:45:17 UTC
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outermeasure wrote:
glebovg wrote:
Let and suppose is an arbitrary large integer. What if, for example, we want to choose s.t. all hold? I cannot use a computer because all integers are arbitrary.

If exists, then the set

exists by axiom scheme of separation (you can write it out in full if you want). No axiom of choice is needed.

On the other hand, to assert is nonempty, you need to dig into the arithmetics of , which again you don't need AC to see that or something like that works.

So, how do I show it is nonempty? How do I show something similar in structure is nonempty? Is there a general approach?

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