# [Agda] termination proofs

Altenkirch Thorsten psztxa at exmail.nottingham.ac.uk
Tue Sep 25 21:30:15 CEST 2012

```The set of Turing machines is countable but we were talking about
*functions*.

The standard proof by diagonalisation that the set of functions from Nat
to Nat is uncountable can be carried out in Agda - this has nothing to do
with being classical.

Assuming that we can find out how a function can be computed (Church's
thesis + choice) is fishy both from a classical and constructive
perspective.

What is difficult to understand about constructive logic? I am having more
trouble with classical logic. Bool = Prop seems a strange idea.

Thorsten

On 25/09/2012 17:33, "Serge D. Mechveliani" <mechvel at botik.ru> wrote:

>On Tue, Sep 25, 2012 at 05:12:01PM +0100, Peter Hancock wrote:
>>
>>> The countability grounds is not relevant here, because I asked about an
>>> _algorithmic_ map in  Nat -> Nat.
>>
>> For me, a function just *is* an algorithmic map which we can recognise
>>as total.
>> The normal Cantor diagonalisation argument shows that Nat->Nat is
>>uncountable.
>>
>> I admit, this will seem strange at first, because there are only
>>countably
>> many Turing machines, and the total ones will be a subset of these.  But
>> the notion of a cardinal is a bit strange constructively.  There are
>> arbitrarily many subsets of a singleton, constructively.  I suggest that
>> this argument is worth a bit more of your attention.
>>
>> (I also admit I am sweeping some subtle points under the carpet.)
>
>I meant classical mathematics and classical logic. According to them,
>the set of total Tuering machines is countable.
>I try now to imagine many subsets of a singleton. And it comes out that
>there are as many as  2^1  of them, that is  2.
>
>I like Agda, despite that understand nothing in a non-classical logic.
>So far I, studied only that proofs are build as members of a dependent
>type, and that negation for T is represented as  T -> \bottom
>(if I do not confuse anything).
>
>Regards,
>
>------
>Sergei
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