Magical Computing

Would it be possible for a wizard to create a universal Turing machine that had an infinite amount of memory?

Also, if we could master magic, do you think that it would be possible to create a separate pocket universe for a computer, that could perform exhaustive searches of an infinite problem space in a way that would seem instant when observed from this universe?

>>13
There exists a set I such that the following properties are true:
! ø ∈ I
! for all x ∈ I: (x ∪ {x}) ∈ I

>>15
infinite sets!

>>15
I've you even such a set and how can "for all x ∈ I: (x ∪ {x}) ∈ I" be true? Can't even imagine such a construct.

>>18
Well it's not constructible using only the finite operations you can perform on sets. Which is why it's an axiom. It's taken to exist.

It's not really a physical construct, it just encapsulates the iterative idea in a set form. Mainly what you do is prove some properties that classes of objects have. Like all even numbers behave in a certain way or something like that. As a programmer, you can think of it as a stream, we make sure never to explicitly calculate elements, but use the definition of the stream itself to do our work.

Sorry that explanation came out terrible and I'm having a hard time thinking of a smart way of saying this.

>>18
I = {ø, {ø}, {{ø}}, {{{ø}}}, ...}.  I believe this is just ℕ as Peano numbers, but I might be wrong, I'm rusty on THEORETICAL SHIT THAT DON'T GET SHIT DONE.

>>21
The definition I gave (standard construction) produces:

I = {ø, {ø}, {ø, {ø}}, {ø, {ø}, {ø, {ø}}}, ...}

You could also do for all x ∈ I: {x} ∈ I, which is what you gave. Generally you just need a successor function S then for each element e in the set, S(e) is also in the set.