It seems that this fundamental particle of set theory needs to be defined then. — keystone
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
I can accept it as an algorithm for generating the set, but not as a completed set — 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". — TonesInDeepFreeze
It's amazing to me that cranks are FULL of criticisms to mathematics but they know nothing about it! — TonesInDeepFreeze
So do you agree that the = in an infinite summation means something different than the = in a finite summation? — keystone
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
I also don't think you want to discuss the intuition further since it's not formal — keystone
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 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
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
I can accept it as an algorithm for generating the set, but not as a completed set....but we've been here before.... — keystone
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
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