On 7 Sep 2012, at 16:13, Gottfried Barrow wrote:
> On 9/7/2012 4:41 AM, Josef Urban wrote:
>> Hi,
>>
>> just a quick comment before this becomes into some HOL vs FOL issue:
>> (conservative) extension by definitions
>> (http://en.wikipedia.org/wiki/Extension_by_definitions) is a standard
>> FOL method that used very often (e.g., when building up math in ZFC).
>> I believe standard logic textbooks (like Mendelson) discuss this
>> early.
>>
>> Best,
>> Josef
>
> ...But before I go any further, I'm open to the idea that I have a basic,
> simple misunderstanding of first-order logic, and that's why I pose the
> simple question, "Can someone please tell me how to define a constant
> for the set which exists which has no elements, and do so using only the
> symbols that have been given to ZFC sets, as ZFC sets has been
> specifically defined and initially locked in according to the general
> FOL specification?"
>
> I looked at Mendelson's book. He may discuss logic extensions somewhere
> in the book similar to what the wiki page describes, but starting on
> page 288, he gives a two page summary of ZF sets. I don't see that his
> summary challenges the traditional view that traditional mathematics
> can, in principle, be reduced down to FOL formulas according to the ZFC
> sets definition, which is based on the general FOL specification.
I think you are referring to the 4th edition of Mendelson's "Introduction to
mathematical logic". I only have the 3rd edition to hand. However, I can see
the contents list of the new edition thanks to Amazon and it is clear that the
book has just grown a bit. The sections you need to read are still there. The 2
pages you quote are in fact at the end of the 62 pages of Chapter 4 of the
book. This chapter, entitled, "Axiomatic Set Theory" undertakes precisely the
programme of building up a working vocabulary of set theory by definitions
starting from a primitive first-order language whose only non-logical symbol is
set-membership. The definition of the null set is on the third page of this
exposition.
The definitional principles used are explained in section 2.9 of the book
entitled "Definitions of new function letters and individual constants".
It so happens that Mendelson chooses the system of set theory called NBG,
because he believes it is better suited to mathematical practice. Hence the two
pages on ZF that you mention are at the end of the chapter in a section
comparing NBG with other systems. The differences between NBG and ZF are
immaterial to your question about how to define the null set. (In fact,
Mendelson gives the definition before giving any of the axioms that are
specific to NBG).
Regards,
Rob.
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