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Topic: Matheology S 224
Replies: 16   Last Post: Apr 21, 2013 6:53 PM

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Posts: 2,710
Registered: 12/13/04
Re: Matheology S 224
Posted: Apr 20, 2013 11:24 AM
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On 20/04/2013 8:59 AM, fom wrote:
> 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.

Right. Fwiw, I had claimed I'd "borrow", say, 'e' and others like '='
as much as I could to express meta level objects (unformalized sets,
individuals, and what not).

There is no remainder in the mathematics of infinity.


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