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user21820
user21820
04:45
@Wildcard If you mean a finite list, then yes. Normally "list" denotes an abstract description of the k-th item on it. For example, the list (0,1,2,3,...) is denoted precisely by the identity function on N, namely ( N k ↦ k ), and there is no natural number greater than all items in this list. I suggest you be a bit more precise when you use a term that already has a conventional meaning. =)
@Wildcard ( S x ↦ E(x) ) is a compact notation for denoting a function that accepts any input x of type S and outputs E(x).
@Wildcard I realize you may object to the complete collection of natural numbers, which is precisely why I didn't use the term "set" anywhere. You cannot however object to the type of natural numbers, namely the ability to recognize any given natural number as indeed a natural number.
Hence you cannot deny the existence of the program g = ( N k ↦ ¬F(k)(k) ), if there is a surjection F from N onto S = Func(N,bool). This condition is not meaningless at all even constructively interpreted, namely it says "F is a program that given an input k of type N will output a program of type S, such that for any object h of type S there is an x of type N for which F(x) = h".
The diagonal argument shows constructively that given any such program F there is a program g that F does not enumerate.
@Wildcard I'm sorry but I do not agree with this. If he had not attempted to derail serious mathematical discussion many times, I would not have bothered. Feel free to ask the moderators about this if you think I'm wrong; I am sure that what I did is acceptable.
Before you judge my judgement, please go and investigate all his shenanigans, so much so that a large portion of active chat users know about it, and he got suspended for a week from Math SE! Furthermore, anyone with sufficient rep should take a look at this and this and this for more evidence.
Wildcard
Wildcard
05:10
@user21820 Yeah, I looked closer after my first comment and was sorry I'd spoken up. :)
user21820
user21820
@Wildcard: Hello! No problem. Let's stick to logic. =)
Wildcard
Wildcard
])
=)
(Just popping in to read your responses, as I really don't have time to reply now...gotta drive in 5.)
user21820
user21820
Sure no problem.
Take your time.
See you around next time! =)
Wildcard
Wildcard
@user21820 Yup!
As I said, I'm enjoying this discussion. Hope you are too!
@user21820 I'm still missing something about your notation here. That arrow with the line beforehand...what?
user21820
user21820
@Wildcard You mean "↦"?
Wildcard
Wildcard
05:13
Yeah
user21820
user21820
It's just the standard symbol called "\mapsto" in LaTeX denoting that what is on the left gets mapped to what is on the right.
Wildcard
Wildcard
@user21820 so you mean a function accepting a boolean and a natural number that returns...no, I don't get it.
Gotta drive now though. I'll pick it up next time. :)
user21820
user21820
Well if you know programming, it's easy to write an actual program to implement the diagonal argument.
If you want, I'll do just that in Javascript.
@Eugene: Hello and welcome! Feel free to ask anything about (mathematical) logic here.
Eugene
Eugene
Thank you @user21820. I heard, though I haven't found evidence yet, that mathematical logic has proven that God does not exist. Can you please tell me the book where the demonstration can be found?
user21820
user21820
@Eugene It is false. Mathematical logic cannot by itself prove anything about the real world, much less about God. However, it lends evidence to the existence of a lawmaker, rather than the opposite.
Eugene
Eugene
05:23
Yes, I know that mathematical logic deals with synthetic a priori truths. But even so, there was supposedly a logician who demonstrated that it is illogical to believe in the existence of God, at least God as we (or many) conceive.
user21820
user21820
Then I need you to more precisely define "God". For example if you define "God" as an entity that can do absolutely anything, then no such God exists.
Eugene
Eugene
God as conceived by Christianity: omniscient, omnipresent, omnipotent.
user21820
user21820
@Eugene The problem is that there are various possible definitions of terms of the form "omni.*".
Take a look at the following posts:
-1
A: Can ancient religions with an omnipotent god be discussed into a definite answer about if the religion is real or not?

user21820If you accept classical logic, then it is a logical fact that there cannot be an absolutely omnipotent entity that can do anything at all, simply by the law of excluded middle. For a specific example: It is impossible (for any entity) to create an uncreated object. Note that this is an abso...

1
A: Is 'strong' omnipotence required to create another omnipotent being

user21820I assume you use "weak omnipotence" for the fourth definition on the Wikipedia article, namely that a weakly omnipotent entity is one that can do anything that is logically possible. In that case I agree that it is not inconsistent that a weakly omnipotent entity can create another weakly omnipot...

Believe me when I say that I give purely logical arguments in those posts. Unfortunately, most people on Phil SE do not have a grasp of basic logic...
@Eugene: You should also take a look at the posts I linked to from them, because they give more detail. I think that after reading them you would more or less have an answer to your inquiry.
Eugene
Eugene
I agree that the meaning of "omnipotent" is problematic. I became acquainted with this problem as I read the process philosophers.
user21820
user21820
So in some sense you are correct to say that the typical (layman) idea of omnipotence leads to a contradictory belief system.
And the wikipedia article gives many weaker notions of omnipotence, some of which can get around the problem.
I personally don't really care, because I've no vested interest in any belief system per se. I believe what I am certain is true, and that's it.
If you are interested I can point you to the reasoning behind my claim that logic lends evidence (not proof) to the existence of a lawmaker.
It does not imply existence of an absolutely omnipotent being, but it is actually the more important issue anyway.

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