All of my work is listed at http://researchmap.jp/seijikoide/?lang=english.

However, it includes papers in English and Japanese, too.

You can get copies of some papers, but cannot others.

If you are interested in some of what you cannot get copies,

please let me know them and your email address.

By the way, you can get my doctor thesis from

http://www.nii.jp/graduate/thesis/pdf/201103/koide_Dr_thesis.pdf

Regards ]]>

Do you have any material from the MOP3 presentation you could forward to me?

Thanks again. Glad to see you’ve still not escaped your metacircle.

]]>The axiom of foundation means that we can find some

In case of

In case of

In case of

So, exactly it should be written as “not a case

In fact, every entity in ZF is a set including the empty set. Therefore, the axiom of foundation is true for this postulate.

If so, we can pick a member that satisfies the axiom of foundation. Then, furthermore we can recursively pick a member that satisfies the axiom from the previously chosen member. Thus, finally we can arrive at the empty set, because we started with a set of finite cardinal numbers, and pick a member of the set. Note that { }∩

Thus, descending chains of membership is terminated at the empty set in ZF.

Oppositely, this axiom is a requirement to inhibit the infinite descending chains of membership.

In case of

If the left side a is {

And I have questions about the first example, that you used to explain the axiom of foundation.

You said that x = {a} is a case in A={{ },{a}}, but not A={a,{a}}. This may be a silly question, but why is this true? Is it because that a set of A={a,{a}} form cannot be constructed?

Or is it, because a ‘set(in this case ‘x’)’ so called ‘foundation of a set’ can only be exist in the set that has a form of {{ },{a}}.

Another question. How about set A={a}? Is this set also does not a case for the x={a} as a foundation too, like the A={a,{a}}?

Please reply. ]]>