> On 14 Mar 2021, at 17:26, Tomas Pales <[email protected]> wrote:
>
>
>
> On Sunday, March 14, 2021 at 10:57:08 AM UTC+1 Bruno Marchal wrote:
>
> But what is an object?
>
> Anything that is identical to itself. It also seems necessary that every
> object is part of a greater object and has properties.
That’s not enough precise. It makes any thing into object, except that it seems
to need a notion of part, and thus can be structured, in many ways, as many as
there are set theories, at least, and there are many, from ZF and NF to the
many toposes...
>
>> We cannot really invoke “reality” as its very nature is part of the inquiry.
>>
>> I regard as reality all objects (that are identical to themselves, of
>> course).
>
> I take x = x as a logical truth about identity. So every thing is equal to
> itself, and so, self-identity cannot be a criteria of (fundamental) existence.
>
> Why not? Why would some objects that are identical to themselves exist and
> other objects that are identical to themselves would not exist? What would
> such an existential distinction even mean?
As a scientist, I try to assume as less as possible, and only things on which
everyone agree.
Most people agree that 4+5=9, and that this implies Ex(x+5 = 9), and do one.
With Mechanism, this is enough, and mandatory (up to a Turing equivalence).
Then, from the numbers’ point of view, they get hallucinated in the difference
between p, []p, []p & p, etc… which will explain the apperance of the laws of
physics, in the mind of large stable collection of universal numbers/machines,
but it would be long to explain this right now. I can give references for more.
>
>
> But the collection of all sets equal to themselves, {x I x = x} is typically
> not a set, despite that collection is equal to itself.
>
> I don't see a difference between collection and set.
A collection is a set in the intuitive sense. A (formal) set, in an axiomatic
theory of set, is an element (in the intuitive sense) of a model of set theory.
Yes, the difference is a bit subtle, but capital when we study the axiomatic of
set theory and its models.
If you identify all collection with (formal) sets, you get contradictions. For
example, by Cantor theorem all set of parts P(S) of a set S is bigger than the
set S: PS > S. But if the collection U of all sets was a set, PU > U, but U is
the set of all sets, so certainly U should be bigger than all sets. And there
are many other contradictions...
> And there is no collection of all collections, just like there is no biggest
> number.
The problem is that there is a collection of all sets, once we define set
axiomatically. We just have to be careful to distinguish set and collections,
once we want use such theories to solve some problem.
>
>
> You seem to assume everything at the start, but without defining things, that
> will lead easily to inconsistencies.
>
> I assume the law of identity for every object, so all inconsistencies are
> thereby ruled out.
? (You seem to assume some object, and it is unclear if you mean “physical
object”, psychological object, etc. You seem to have a general theory of
object, but you don’t seem to characterise them axiomatically, so, to be
honest, I don’t see any theory.
>
> A square circle is equal to itself, arguably.
>
> No, a square circle is a circle that is not a circle,
Obviously, a square circle is a circle. It is a counter-example to the idea
that a square cannot be a circle. It does not exist, but that does not make it
different to itself. Even a Unicorn is equal to itself.
The real problem of the square circle is that we don’t have provides
definition, or we talk about something which cannot exist, so we can’t really
build meaningful proposition about it, without being inconsistent.
> so it is not identical to itself. It is not an object, it's nothing.
x = x is usually an axiom of all theory of identity, with the symmetry and
transitivity. At least we agree on that. Your notion of object is still too
much fuzzy to be used in metaphysics/theology, Imo.
Bruno
>
>
>
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