On 9/6/2012 11:09 AM, Brian Tenneson wrote:
Correct me if I'm wrong but my understanding is that sets and membership cannot be defined in terms of a more primary mathematical concept. Functions can be defined in terms of this primitive called sets. Numbers are sets; natural numbers are defined directly in terms of sets (via the Von Neumann approach) and every more complicated number set can be defined in terms of the previous type of number set all the way up to real numbers, complex numbers, and nonstandard number sets. The only type of number I am not sure how they can be seen as sets is that of surreal numbers described by Conway I believe. I don't know much about surreal numbers.


Yes, this approach necessitates the existence of sets and membership.
Hi Brian,

Surreals and hyperreals and non-standard numbers and so on, the list is long! My point is that there really is no such thing as an absolutely irreducible entity.


There probably are other ways to define numbers but all properties that we want numbers to have can come from how they are defined in terms of sets. In other words, the set theoretical description of various number sets is sufficient.


    Depending on what intends to try to explain, but sure.

Kronecker said "God made the integers; all else is the work of man." I would amend that to say God made sets (and membership); all else is the work of man.

Balderdash! We can use the God concept as a way to capture the sum of what exists and its evolution and so forth, but it is just another word that may not refer to anything that really exists.


On Thu, Sep 6, 2012 at 7:45 AM, Stephen P. King <stephe...@charter.net <mailto:stephe...@charter.net>> wrote:

    Dear Brian,

        "can be defined ..." implies the necessary existence of
    something or process or whatever that does the act of defining the
    set. Truth values do not do this, btw. Sets are collections
    defined in terms of functions, but numbers in-themselves are not
    those functions.. Unless you are considering some other ideas of
    what sets are... If we are going to think of set as having
    ontological primacy we have to have a notion of a set that does
    not need a membership function.



    On 9/6/2012 10:28 AM, Brian Tenneson wrote:
    All numbers can be defined in terms of sets.  The question
    becomes this:
    do sets have ontological primacy relative to mankind or are sets
    invented or created by mankind?

    On Thu, Sep 6, 2012 at 5:11 AM, Roger Clough <rclo...@verizon.net
    <mailto:rclo...@verizon.net>> wrote:

        Hi Stephen P. King
        Yes, of course, but I wanted a more obvious, dramatic example.
        The philosophy of mathematics says something like the numbers
        belong to a static or eternal world, change itself  is a
        property of geometry.
        Numbers and geometry thus belong to the platonic world,
        which is forbidden or at least not consistent with the philosophy
        of materialism, IMHO.
        If numbers are platonic, I wonder what the  presumably
        materialist
        Steven Hawkings has to say about their origin in his recent
        book on numbers.
        Roger Clough, rclo...@verizon.net <mailto:rclo...@verizon.net>
        9/6/2012
        Leibniz would say, "If there's no God, we'd have to invent him
        so that everything could function."

            ----- Receiving the following content -----
            *From:* Stephen P. King <mailto:stephe...@charter.net>
            *Receiver:* everything-list
            <mailto:everything-list@googlegroups.com>
            *Time:* 2012-09-06, 07:53:18
            *Subject:* Re: Could we have invented the prime numbers ?

            Dear Roger,

                Could the mere possibility of being a number (without
            the specificity of which one) be considered to be "there
            from the beginning"?




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