Math Forum Discussions

fom

Posts: 1,968
Registered: 12/4/12

Re: Matheology S 224
Posted: Apr 20, 2013 10:59 AM

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On 4/20/2013 5:25 AM, Alan Smaill wrote:
> Frederick Williams <freddywilliams@btinternet.com> writes:
>
>> Nam Nguyen wrote:
>>>
>>> On 19/04/2013 5:55 AM, Frederick Williams wrote:
>>>> Nam Nguyen wrote:
>>>>>
>>>>> On 18/04/2013 7:19 AM, Frederick Williams wrote:
>>
>>>
>>>>
>>>>>> Also, as I remarked elsewhere, "x e S' /\ Ay[ y e S' -> y e S]" doesn't
>>>>>> express "x is in a non-empty subset of S".
>>>>>
>>>>> Why?
>>>>
>>>> It says that x is in S' and S' is a subset of S.
>>>
>>> How does that contradict that it would express "x is in a non-empty
>>> subset of S", in this context where we'd borrow the expressibility
>>> of L(ZF) as much as we could, as I had alluded before?
>>
>> You really are plumbing the depths. To express that x is non-empty you
>> have to say that something is in x, not that x is in something.
>
> but the claim was that x *is in* a non-empty set --
> in this case S', which is non-empty, since x is an element of S',
> and S' is a subset of S.
>
> (Much though it would be good for Nam to realise that
> some background set theory axioms would be kind of useful here)
>

Yes. I thought about posting some links indicating
that primitive symbols are undefined outside of a
system of axioms (definition-in-use)

The other aspect, though, is that Nam appears to be using an
implicit existence assumption. So,

AxASES'(xeS' /\ Ay(yeS' -> yeS))

clarifies the statement and exhibits its second-order nature.
This is fine since he claims that his work is not in the
object language.