@user170039 I don't agree with you trashing this message, which explains a quote from the same notes by Asaf; it was relevant to your question about an existing logician who has a similar view as the one I've been explaining so far, and the quote you trashed explicates Asaf's view about natural numbers in the real world. So I'll post it again and please do not trash it.

If it's not clear enough, Asaf states not only that (in his view) mathematics is not intended to match reality (and if so then in my view not philosophically sound), but intriguingly also claims that in reality there is no infinite object/set, nothing larger than 2^(10^11), no fractions or irrationals, no negative numbers, but there is 1,2,3,... up to some n < 2^(10^11). This claim clearly mirrors my claim that PA is at least at small scales a good approximation of some entity in reality!

@user170039: I understood that. But you also trashed another comment.

I only reinstated that other comment.

@user170039: Oops! Sorry I didn't realize that.

I'll trash the rubbish I created just now.

← 2 messages moved from Philosophy of Mathematics

It doesn't matter.

So why do you repeat what I said?

@user170039 And by the way, if you want to know what I think is crucially important, it's not mathematics or philosophy. It's morality.

@user170039 You are either attempting to troll me, or you are just very very naive.

@user170039 It's relevant because you mentioned what is crucial to you. So I mentioned what is crucial to me.

Let's just say I think we're better off not talking about philosophy anymore until you've studied basic proof theory and model theory. At least then I wouldn't be spending unnecessary time when my answers to your question involve technical details that you don't yet grasp. So yes, let's just stop for now.

@user170039: I would love to hear your views again after that. =)

Suppose that $\beta$ is constructable and has a CF of $\Omega_x$ with $x>0$. Then the largest $\zeta<\beta\land \zeta\in C(\alpha,\beta)_n$ must replace $\Omega_\alpha$ with $\psi(x,y)$, where $x<\alpha$ and $y<\beta$

@user170039 I'm not surprised that I cannot convince people who do not have the necessary knowledge. Have you heard of the Dunning Kruger effect?

It's not about you, but it describes an effect where sometimes people who don't know something don't realize they need to know it before they can make accurate judgements.

$y<\beta$ must hold true because we cannot construct $\psi(\gamma,\zeta)$ with $\zeta\ge\beta$

You are also saying that Asaf's position is inferior by the way. I'll leave you to believe what you want for now.

← 2 messages moved from This is the Realm of Simply Beautiful Art

If you do not eventually understand, I suppose I have in fact wasted hours of my time on you.

← 1 message moved from Philosophy of Mathematics

For simplicity,
x-1 → y,          if x = y+1
x-1 → x[0,n],     if x is a limit and is CCF
x-1 → {0|x[p,~]}, if x is a limit and isn't CCF

We also say that anything of the form {x|y} is CCF.

(x+y)[p,q] → (x+(y[p,q]))
(x*y)[p,q] → (x*(y[p,q]))
(x^y)[p,q] → (x^(y[p,q])
{x|y}[0,n] → y applied n times with initial value of x
Ω(0) = {0|~+1}
Ω(x)[p,q] = ψ(p,q), x>0
ψ(a,0) = {0|Ω(a)^~}
ψ(a,b) = {0|ψ(a,b-1)^~}

← 1 message moved from This is the Realm of Simply Beautiful Art