Wrong. A series is a certain kind of function. The continuum is not a function. Wrong. I explained the difference between them. Knowing the definition...
You're still conflating 'continuum' with 'continuous'. They are closely related concepts, but not the same concept. Also, the least upper bound proper...
What does that mean? Ordinarily, "to index" means to make a set the range of a function, as the domain is the index set. The domain is the indexing se...
My point was that he didn't ask for a definition of 'the continuum'. The takeaway for you is to not conflate 'the continuum' with 'continuous'. I didn...
The speed of Achilles is 10meters/1second. The speed of Tortoise is 1meter/1000seconds. I applied mathematics to determine that, in the 100 meter race...
Who is "we"? Other than you? What does "between" mean for sets in general? Or do you mean for real intervals? Usually, 'continuous' refers to function...
He didn't ask for a definition of 'the continuum'. 'the continuum' is a noun. He asked for the distinction between 'continuous' and 'discrete'. 'conti...
I am not a mathematician. Proof: Let Mx <-> x is a mathematician Let Rx <-> x produces results in mathematics Let t = TonesInDeepFreeze 1. Ax(Mx -> Rx...
I have a good grasp of the some of the basics of set theory, but I am not very knowledgeable beyond those basics. Anyway, the idea of someone, who doe...
No one said anything about ZF. But if it is taken that there only finitely many things in what is designated as 'the real world', and it is regarded t...
(1) In open forums like this, there is usually more disinformation and confusion about mathematics than there is information and clarity. Instead, pro...
That you have limited time for mathematics is all the more reason for not wasting that limited time in routes that lead to dead ends, misinformation a...
What 1:1 map are you referring to? A 1:1 map from a real interval into points in space? A 1:1 map from a real interval into points of time? A 1:1 map ...
Please please please do not refer to AI bots for math proofs. They are so often incorrect. I've tried it a few times, and the bot gives clearly incorr...
Start with one thing: The poster asked about a "map into" as a "1:1 relation between". A word for that is 'injection'. That's fine with me. And if you...
First, again, I don't know what the poster means by "the real world" so I don't know what firm and clear notion there is of an injection from the set ...
I don't opine here on that other question. But what do you mean by "maps into"? Do you mean "there is a function into" or do you mean "there is a one-...
I'm keeping to first order set theory. The point is that in set theory, any two complete order fields are isomorphic. You can think of that as saying ...
Yes. I said it many posts ago in a post you claimed to have read. Set theory doesn't provide a definition of 'is a number'. Rather, set theory provide...
fishfry has been giving you the info. But to understand very well, you need to go in forward direction from page 1 of a book - from the simplest conce...
Your questions arise only because you don't know anything about the subject, and also because (contrary to your claims) you don't carefully read the a...
Fine. But that's a different domain than the one I used. Yes, it's a common fact. The set of natural numbers is aleph_0. Every set is one-to-one with ...
It's not counterintuitive that there exist functions whose domain is the set of positive natural numbers. Any Calculus 1 textbook has such functions a...
They are almost entirely mathematics, very little philosophy. Except the introductory chapter in 'Introduction To Mathematical Logic' by Church. A pre...
I didn't use the word 'infinity' in that post. I used 'inf' (usually seen as the lemniscate). In the context and sense I used, 'inf' has no meaning by...
That you ask that question, indicates that you don't know what 'aleph' and 'countable mean' despite that you toss those terms around as if you're maki...
Then say 'aleph_0' since, in this context, 'infinity' is ambiguous. You think that only if you didn't read my post. For the third time: The domain of ...
An ordinary route: Prove the existence of a system of naturals, and define a particular system of naturals. Prove the existence of a system of integer...
As you touch on, the system of naturals can be extended to a system of rationals, and the system of rationals can be extended to the system of reals. ...
What do you mean by "infinity" used as a noun? There is the adjective "is infinite": S is infinite if and only if S is not finite. And there are vario...
I have the first edition in print. I don't have an errata sheet for it. But I do have pencil marks for the errata I caught. There is an errata sheet f...
Many posts ago I addressed the fact that I am naming an expression and so the definiens is in quotes. For about the 20th time, as you skip each time, ...
Don't take my word for it. Look up 'non-standard analysis', 'hyperreal', 'infinitesimal', 'internal set theory'. Better, read Enderton's beautifully w...
That is blatantly false and with an exclamation mark that is a cherry on top of falsehood. I wrote: This latest time I just made that proof even more ...
An operation symbol takes only finitely many arguments. For example, the operation of division (x/y) takes only two arguments (x and y), and is define...
"The Pentastring is this sentence has five words" No one has anything to say about any "content" of that other than that it's gibberish. These are all...
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