> On 2 Oct 2018, at 10:14, agrayson2...@gmail.com wrote:
>
>
>
> On Tuesday, October 2, 2018 at 7:20:10 AM UTC, Bruno Marchal wrote:
>
>> On 1 Oct 2018, at 14:20, agrays...@gmail.com <javascript:> wrote:
>>
>>
>>
>> On Monday, October 1, 2018 at 11:47:47 AM UTC, Bruno Marchal wrote:
>>
>>> On 30 Sep 2018, at 16:30, Philip Thrift <cloud...@gmail.com <>> wrote:
>>>
>>>
>>>
>>> On Sunday, September 30, 2018 at 4:50:01 AM UTC-5, Bruno Marchal wrote:
>>> [Re:] forcing theory in set theories with classes.
>>>
>>>
>>> Bruno
>>>
>>>
>>>
>>> Do you follow the work of Joel David Hamkins (forcing applied to
>>> set-theoretic "multiverse", etc.)
>>>
>>> (I have a basic idea of a type-theoretic parallel to this.)
>>>
>>> The set-theoretic multiverse
>>>
>>> https://arxiv.org/abs/1108.4223 <https://arxiv.org/abs/1108.4223>
>>>
>>> Joel David Hamkins
>>> @JDHamkins
>>> Professor of Logic, University of Oxford, and Sir Peter Strawson Fellow in
>>> Philosophy, University College Oxford. Formerly of New York.
>>> http://jdh.hamkins.org <http://jdh.hamkins.org/>
>>>
>>
>> The math is interesting, and could be of some use, but it is a priori far
>> too much Aristotelian to be coherent with the mechanist hypothesis. That
>> should follow “easily” from the result described in most of my papers on
>> this subject. The author does not seem aware of the mind-body problem, which
>> put extreme constraints on what the physical reality can come from. Even
>> Peano arithmetic, although integral part of the notion of observer, is too
>> much rich for the ontology, where not only the axiom of infinity is too
>> strong,
>>
>> Since you want to banish the concept of infinity from mathematics, how would
>> you define, say, the limit of an "infinite" series? How would you even
>> discuss this series in the context of finite mathematics? AG
>
>
> Good question.
>
> The answer is not simple technically. The point is that using only the theory
> Q (Robinson Arithmetic) or SK (the combinators), I can define the universal
> (Turing, Church) machine, and the concept of infinity will be a tool used by
> them in their mathematics.
>
> I do not ban anything from mathematics, nor from physics. I ban only infinity
> from the ontological terms. I ban only infinity in the metaphysics/theology.
> (Even God is not ontological, like in Proclus or Plotinus theology).
>
> Have you understand the post on Church’s thesis. You might tell me as this
> will help me to see how to proceed to make you grasp all this.
>
> Bruno
>
> You only ban infinity from ontological terms? I have no idea what this means.
It means that 0 exist, 1, exists, 2 exists, etc.
> I do know you start with the natural numbers, presumably an infinite set and
> existing in some Platonic realm.
Not really. Only 0, 1, 2, …
But not {0, 1, 2, 3 …}, which is not a natural number.
It means that my axioms, for the whole theory of everything including
consciousness is literally just classical logic + the axioms of Q:
1) 0 ≠ s(x)
2) x ≠ y -> s(x) ≠ s(y)
3) x ≠ 0 -> Ey(x = s(y))
4) x+0 = x
5) x+s(y) = s(x+y)
6) x*0=0
7) x*s(y)=(x*y)+x
There is no infinity axiom, nor any infinite object in the intended model. What
is proved from Q is true in all models (interpretations) of Q.
I don’t even allow the induction axioms, despite the phenomenology use them, as
Q is rich enough to mimic the believer in the induction axioms, and indeed the
believer in infinity (like the ZF machine).
To grasp this it is important to understand the difference between compute and
proof.
Keep in mind that Q can mimic ZF proving the consistence of Q; but that cannot
convince Q of its consistency (by the second incompleteness theorem of Gödel).
> So I have no idea about your aversion or denial of infinity.
No aversion at all. It is just part of the phenomenology, and if I put it in
the ontology, the “white rabbits” becomes to numerous, and the physics predicts
too many things.
> As for the Church's thesis, I have set aside a copy of Chrome with several
> relevant topics which I see as prerequisites to that understanding including,
> for example, Cantor's theorem, but have yet to get into it seriously due to
> personal issues and computer problems in Russia and Ukraine (the latter now
> solved). But when I do, I'll get back to you. AG
OK. Normally my post was self contained. (Except for the notion of function).
Ask any question.
Bruno
>
>
>
>
>
>>
>> but even the induction axioms are too strong.
>>
>> Pragmatically, sets and typed lambda terms or typed combinators can indeed
>> be very useful.
>>
>> Bruno
>>
>>
>>
>>
>>
>>
>>
>>>
>>> - pt
>>>
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