It seems that this fundamental particle of set theory needs to be defined then. — keystone

From the axioms, we prove that there is a unique x such that x has no members:

E!xAy ~y e x

Then we define :

0 = x <-> Ay ~ y e x.

And, informally, we nickname that "the empty set".

WRONG. Typical claim of someone who knows not even the first week of Calculus 1.

An infinite summation is a LIMIT, not a final term in the sequence, as the sequence has no final term. — TonesInDeepFreeze

So do you agree that the = in an infinite summation means something different than the = in a finite summation?

I can accept it as an algorithm for generating the set, but not as a completed set — keystone

It's not an algorithm. It's not "generating" a set. It is a certain tuple.

It's amazing to me that cranks are FULL of criticisms to mathematics but they know nothing about it!

From the axioms, we prove that there is a unique x such that x has no members:

E!xAy ~y e x

Then we define :

0 = x <-> Ay ~ y e x.

And, informally, we nickname that "the empty set". — TonesInDeepFreeze

I'm going to let you have the last word on this point for now since I'm trying to keep the discussion focused on Thomson's lamp and continuous objects vs. points.

It's amazing to me that cranks are FULL of criticisms to mathematics but they know nothing about it! — TonesInDeepFreeze

Look at the history, I've tried to end the discussion on this point but you keep looping it back it!

So do you agree that the = in an infinite summation means something different than the = in a finite summation? — keystone

Absolutely I do not agree.

'=' stands for equality. Period.

So, now I have to give you a free lesson from the first chapter of Calculus 1.

The infinite sum here is:

Let f(0) = 1. Let f(k+1) = f(k)/2

df. SUM[k e N] f(k) = the lim of f(k) [k e N]

thm. the lim of f(k) [k e N] = 2

/

Did you not take Calculus 1 when you were in school?

I'm going to let you have the last word on this point for now since I'm trying to keep the discussion focused on Thomson's lamp and continuous objects vs. points. — keystone

Well, you asked me.

And, very likely you'll have the last word anyway, since I really am out of time for a while.

I also don't think you want to discuss the intuition further since it's not formal — keystone

Wrong and contradicts what I've said about that, and contradicts the fact that I (unlike you) DO read in the philosophy of mathematics about different intuitions, including the range of finitism. And I( (unlike you) DO read about formalizations of those different intuitions.

I have nothing against expressing intuitions. But you do more than that. You compare your intuitions with set theory, including criticism of set theory, but you don't know anything about set theory so your criticisms are woefully ill-premised. That's where I come in and say, "Whoa whoa whoa there, pardner, it's fine to have criticisms of set theory, but you better know what you're talking about."

Look at the history, I've tried to end the discussion on this point but you keep looping it back it! — keystone

What in the world? Your comment about Peano systems was ludicrously ignorant and so wrong it's not even coherent. So I corrected you. There's no "looping back" by me.

What in the world? Your comment about Peano systems is ludicrously ignorant. So I corrected you. There's no "looping back" by me. — TonesInDeepFreeze

I'm not in a position to argue that Peano systems are inconsistent so I'd like to set this aside for now. — keystone

Absolutely I do not agree.

'=' stands for equality. Period.

So, now I have to give you a free lesson from the first chapter of Calculus 1.

The infinite sum here is:

Let f(0) = 1. Let f(k+1) = f(k)/2

df. SUM[k e N] f(k) = the lim of f(k) [k e N]

thm. the lim of f(k) [k e N] = 2 — TonesInDeepFreeze

I have developed a paradox which (I think) shows the problem of using limits to define equality. Let's shelve this discussion until then. If you're up for it, I'll hold off on submitting a new discussion on that paradox until you're back as I'm sure you'll have an interesting opinion. Of course, I'm not expecting for any commitment from you whatsoever. It's a free world...this is just a nod to you, noting that I'd benefit in you seeing my post....

What in the world? Your comment about Peano systems is ludicrously ignorant. So I corrected you. There's no "looping back" by me.
— TonesInDeepFreeze

I'm not in a position to argue that Peano systems are inconsistent so I'd like to set this aside for now.
— keystone — keystone

Yet you went on to ignorantly argue about it!:

I can accept it as an algorithm for generating the set, but not as a completed set....but we've been here before.... — keystone

You don't get say that you don't want to argue about it, then argue about it anyway, then blame your interlocutor from replying. Not on this planet at least.

Hey, I'm all for you dropping whatever you want to drop. Saves me from cleaning up your messes. But "setting aside" doesn't Orwellian mean "I continue to remark on it, then say I want to set it aside, then if you reply you're "looping" back"."

Yet you went on to ignorantly argue about it! — TonesInDeepFreeze

Fair criticism

I can't help but think that the current "rigorous systemization of mathematics for the science that does not abide by thesis C" is inconsistent. I cannot prove that formally, but I can discuss the infinity paradoxes. — keystone

Here is my careful philosophical response: Bull.

All you did there is argument by mere repeated assertion, as you ignored the rebuttals I've already given.