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Topic: The set of natural numbers
Replies: 8   Last Post: Oct 1, 2017 5:40 PM

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Alan Smaill

Posts: 1,050
Registered: 1/29/05
Re: The set of natural numbers
Posted: Sep 27, 2017 5:39 AM
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mitch <mitch@dev.null> writes:

> On 09/22/2017 06:37 AM, Alan Smaill wrote:
>> "Julio P. Di Egidio" <julio@diegidio.name> writes:
>>

>>> On 21/09/2017 19:03, Jim Burns wrote:
[...]
>>>> So, I am ready to believe that this lecture series was linked to
>>>> by Julio Di Egidio, but I don't see (yet) why he linked to them.

>>>
>>> I must apologies, I cannot find that post myself, which means I must
>>> have posted it to some other thread: which now I cannot find, either.
>>>
>>> I had actually linked to a specific time within Lecture 1, where
>>> Prof. Schuller utters the magic words "pre-mathematical numbers",
>>> indeed just to help clarify what I am talking about and the kind of
>>> problems I am trying to discuss:
>>> <https://youtu.be/V49i_LM8B0E?t=57m19s>

>>
>> From my fallible memory, maybe another time, you pointed out his comment
>> at the end of the second lecture, following on the axiom of foundation,
>> where he says roughly that every set needs to be built out of only the empty
>> set. IIRC you thought that that poses some problem where infinite sets
>> are concerned.
>>
>> This looks as though you think this is somehow linked with "infinitary"
>> theories in general??
>>

>
> But, is it not?


First, there are different possible meanings for "infinitary" here;
I'm not sure what is intended by Julio di Egidio or yourself
or indeed Jim Burns.

At any event, the place this appears in the lecture is after introducing
the axiom of foundation, which comes after all the other axioms.

My understanding of the claim that every set is built out of only the
empty set is that, if we envisage sets as trees based on the
membersip/epsilon relation, then all the leaves are necessarily the
empty set. (There are no infinite branches, but there can be infinitely
many child nodes of a given node.)

> In general, any existence axiom declaring a set M to
> be an existent will admit the use of
>
> { x in M : x =/= x }
>
> against the
>
> Ax( x = x )
>
> derivable from the open axiom schema of reflexive
> identities from first-order logic. By separation,
> you obtain the empty set.
>
> The statement about being built from only the empty
> set is actually a reference to the situation where
> what exists in the theory exists in the cumulative
> hierarchy.


Definitely.

> If one takes the statement about the
> empty set literally, one is constrained to consider
> only
>
> ZF - Inf


Without Inf we only have ways of proving existence of
finite sets, I agree.

> And, since axioms of infinity are fundamentally
> self-justifying, this would be constructively the
> same as
>
> ZF - Inf + ~Inf


I don't follow the logic here;
ZF - Inf is agnostic about the existence of infinite sets.
Isn't it possible in a pre-theoretic sense to be agnostic
on that question?

> In order to have infinite sets, one requires the
> existence of one infinite set and some form of
> replacement.


These are around in the lecture at the point the comment is made.

> mitch


--
Alan Smaill





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