The set of all sets exists.

[Skepdick]

So in order to actually see the set of all sets you have to stop being a literalist for a bit.

You can't see something that does not exist. And if you're talking about something that does exist, then you're merely calling something else the same name as something that does not exist.

That's not true.

3rd option. You are giving a name to something which does not exist.

Like 0.

[Skepdick]

For poetic effect...

Axiom 1: S is the set of all sets

False premise. No such set exists.

That's not really true. But in order to actually see it, one has to stop being a literalist for a bit.

[Magnus Anderson]

As you wish. I am not going to be drilling inside your skull.

[Skepdick]

As you wish. I am not going to be drilling inside your skull.

You can't even drill out of literature in order to stop being a literalist.

[godelian]

He axiomitizes S into existence. I am actually not particularly critical about this particular custom axiom.

You can't "axiomitize" things into existence.

If something does not exist, it does not exist. You can't bring it into existence.

Well, all axioms axiomatize things into existence. They are the only reason why things exist in the theory.

[Magnus Anderson]

Well, all axioms axiomatize things into existence. They are the only reason why things exist in the theory.

Not quite. That's not what axioms are. In mathematics, something is said to exist if it has the absolute minimum possibility of existing in reality. It does not have to exist in reality. Square-circles, for example, are said to not exist, not merely because they are nowhere to be found in reality, but because they are contradictions in terms, something that has zero possibility of existence. People like Skepdick try to "axiomitize" such things into existence by redefining words. But you can't make X exist by calling something else, something that actually exists, the same name as X.

[Skepdick]

Not quite. That's not what axioms are. In mathematics, something is said to exist if it has the absolute minimum possibility of existing in reality.

Bullshit. Show me any euclidian/2-dimensional object in reality.

It does not have to exist in reality. Square-circles, for example, are said to not exist, not merely because they are nowhere to be found in reality, but because they are contradictions in terms, something that has zero possibility of existence. People like Skepdick try to "axiomitize" such things into existence by redefining words. But you can't make X exist by calling something else, something that actually exists, the same name as X.

This is precisely why you keep missing the point.

A proof by contradiction means only that the object cannot exist within a given axiom-system.
You are 100% correct that square circles can't exist within Euclidian geometry.

To go and say that square circles cannot exist in general is a hasty generalization fallacy.

It simply means you don't know how to conceive of; and define square circles into existence. You lack imagination. That's all.

Absolute impossibilities don't exist in a domain without a priori constraints constraints. It's a blank slate - you make the rules.

[Magnus Anderson]

Show me any euclidian/2-dimensional object in reality.

No need for that. The mathematical concept of existence is different from the physical one. A difference that I've already explained. If you can't understand it, and you're not willing to listen, there is nothing I can do to help you. Sorry.

A proof by contradiction means only that the object cannot exist within a given axiom-system.
You are 100% correct that square circles can't exist within Euclidian geometry.

To go and say that square circles cannot exist in general is a hasty generalization fallacy.

You're clueless, Skeppie.

A proof by contradiction means that the object has no possibility of existence whatsoever.

Square-circles cannot exist at all.

That you can redefine the term "square-circle" so that it means something else is no argument at all.

And the fact that you can't understand that after so many people have tried to help you understand it simply means you're extremely unintelligent.

It simply means you don't know how to conceive of; and define square circles into existence. You lack imagination. That's all.

It doesn't take much imagination to redefine words in such a way so that you can call a diamond a square-circle.

The extremely dumb "You just lack imagination" argument is just that -- extremely dumb.

[Skepdick]

No need for that. The mathematical concept of existence is different from the physical one.

Great! So you agree.

What constrains mathematical non-existence?
What prevents a mathematical object from existing?

A proof by contradiction means that the object has no possibility of existence whatsoever.

Square-circles cannot exist at all.

That's simply not true.

Proof by contradiction shows impossibility within a given logical system
It doesn't demonstrate absolute impossibility

That you can redefine the term "square-circle" so that it means something else is no argument at all.

It's not that I can redefine them. It's that you can't well-define square; and circle canonically and then prove their mutual exclusivity.

You lack universal, context-independent definitions for squares; and circles. You simply have no idea what you are talking about when you use those words.

[Magnus Anderson]

It doesn't demonstrate absolute impossibility

You wish.

[Magnus Anderson]

You simply have no idea what you are talking about when you use those words.

Tsk. It is YOU who do not understand what these terms actually mean.

[Magnus Anderson]

One only has to take into account that you do not even understand what the word "unicorn" means.

Extreme lack of intelligence.

[Skepdick]

You simply have no idea what you are talking about when you use those words.

Tsk. It is YOU who do not understand what these terms actually mean.

I understand just fine how to use them informally and imprecisely.

What I don't understand is why you think they are mutually exclusive.

[godelian]

Well, all axioms axiomatize things into existence. They are the only reason why things exist in the theory.

Square-circles, for example, are said to not exist, not merely because they are nowhere to be found in reality, but because they are contradictions in terms, something that has zero possibility of existence. People like Skepdick try to "axiomitize" such things into existence by redefining words. But you can't make X exist by calling something else, something that actually exists, the same name as X.

Concerning S, the proper class of all sets, I do not see why its existence would always be contradictory.

The existence of S does indeed potentially fly in the face of ZFC's axiom of regularity. However, by insisting that it is a proper class, S could actually be viable. In my opinion, a very carefully crafted concept of "proper class of all sets" may actually be consistent. I do not believe that S is completely outlandish.

However, the current proposal, in its current incarnation, still causes a raging forest fire in arithmetic theory.

As mentioned previously, designated fragments of set theory and arithmetic theory are bi-interpretable. The three axioms in the proposal do not succeed in breaking this connection in any way.

The third axiom proposed forces a solution into existence for the equation n=2^n, while arithmetic (PA) inductively proves over the natural numbers that such solution is simply not possible. That is why, in my opinion, the proposal is not viable in its current incarnation.

[Magnus Anderson]

Concerning S, the proper class of all sets, I do not see why its existence would always be contradictory.

Due to the definitions of the terms involved.

The word "all" indicates that the set S also contains itself.

But a set cannot contain itself.