Cantor's absolute infinity was the collection of all ordinal numbers, which he called an "inconsistent multiplicity," since he recognized that the col...
Cantor thought the absolute infinity was God. I don't know if he ever claimed it was the whole story of mathematics. Don't recall reading whether Göde...
This is what I'm confused about with your objection to the extended reals (or integers, etc.) I noted that the extended reals are essentially a notati...
At this point I'm just trolling you. You're sometimes an easy target because you take yourself so seriously; and I was born a wiseass and can't help m...
Was it not perfectly clear the other day that the usual order on the integers carries over to the extended integers? Where was your reasonable liberty...
"During a lecture, the Oxford linguistic philosopher J. L. Austin made the claim that although a double negative in English implies a positive meaning...
"When caught in a material error, I just claim I didn't really mean it that way." Too tedious to likewise mock and debunk every single point you made ...
It's not any term of the sequence. It's the LIMIT of the sequence. I know you took calculus a long time ago. They taught you that 0 is the LIMIT of th...
I don't either. I agreed with you. I originally said that the extended reals are essentially a notational convenience. As are the names of the numbers...
I did say that the extended reals are essentially a notational convenience. Technically we could live without them. But why? By that reasoning we shou...
Yes. https://en.wikipedia.org/wiki/Extended_real_number_line The extended reals are used so that we can write things like \displaystyle \lim_{x \to \i...
As a complex analysis guy you use the hypothetical point at infinity of the Riemann sphere all the time, don't you? It's just the one-point compactifi...
It's a pedagogical point. It's far far easier for people to understand the point at infinity as the index of the limit of a sequence, than to explain ...
Consider the following set \mathbb R \cup \{-\infty\} \cup \{\infty\} \setminus (\mathbb R \setminus \mathbb Z) with the order -\infty < n \ < \infty ...
If you believe in the extended reals, just setminus all the finite non-integers and you're left with the integers along with the two points at infinit...
Yes exactly. The geometric idea is exactly the same. Consider the half-open unit interval \cup \cup \cup \cup ... That's a geometric decomposition of ...
Do you know what a limit is? The sequence 1/2, 1/4, 1/8, ... has 0 as a limit. If you read through the supertask thread that's been referenced elsewhe...
1, 2, 3, 4, 5, 6, ... Is that not an infinite sequence? (You mean sequence. A series is a sum) It it not indexed by the natural numbers? Take the sequ...
Correct meaning you understand that the rationals are dense but not continuous? Haven't we been doing that all along? Not sure what you mean. The rati...
Are the rationals continuous? Between any two rationals there is another rational, right? But the rationals are full of holes. For example the set of ...
I did. I said that if there are infinitely many things in the world AND that ZF applies to them, then questions of higher set theory become subject to...
Ok. Injection it is. The rest of my point holds. If there is an infinite collection of anything in the physical world; and if ZF applies to the infini...
The world that is studied by physics. The phenomena around us that are amenable to experiments. Things that have mass, electric charge, velocities, an...
The Democrats used to be the party of the working class. They've become the party of the wealthy liberal elites and the poor who benefit from governme...
Here's my (meta-)argument. If the real numbers are instantiated in the real world, then questions such as the axiom of choice and the Continuum hypoth...
A set is defined to have cardinality \aleph_0 if it can be bijected to the natural numbers. Clearly the natural numbers can be bijected to the natural...
It's not. Recall the Cantor diagonal argument, which shows that the cardinality of the reals is strictly greater than the cardinality of the naturals....
That's literally the definition of \aleph_0. A set is defined to have cardinality \aleph_0 if the set can be placed into bijective correspondence with...
Let's look at some simpler examples. 1) Consider the set \{3, 18, 58, 334\}. That's a set with 4 elements, yet 4 is NOT an element of the set. 2) Cons...
Pretty tight in both houses. In general, the US economy does better when the opposition party controls Congress. The government can get into less misc...
Sure. If you had a set \{1, 2, 47, \aleph_0\}, that would be a set with four elements, one of which is \aleph_0. Then if you had a function f defined ...
Uh-oh. We had a very lengthy thread about supertask a while back. Best leave that one alone :-) Well math is math and physics is physics. Math is a to...
It's a general theme of mine that environmental do-gooding generally results in disaster and misery. Your response was interesting. Clearly you're get...
The continuum is a mathematical abstraction. It has no representation or instantiation (as far as we know) to anything in the physical world. This is ...
There are limits. As an example, consider the sequence 1/2, 1/4, 1/8, 1/16, ... We all know that the limit of this sequence is 0. You can certainly ca...
No worries, I no longer remember. I did happen to run across something yesterday. The British government put out a big report on the Grenfell disaster...
I'm beyond explaining this. Let's agree to disagree. Eminently sensible and moderate. You're trolling me now. I'm kind of done here. I can always tell...
This is incoherent at best, wrong at worst. I explained this to you at length. But look. You are trying to prove there are two points without a third ...
I'm not sure what @"sime" meant by that statement either. But ultrafilters are just a set theory gadget that lets you rigorously construct the hyperre...
Recent developments in the West are very concerning. Robert Reich, Clinton's Secretary of Labor, just called for "reining in" Elon Musk. https://www.t...
You are agreeing that free speech is a virtue then. Yet you don't seem too bothered by the globalist war on free speech. It's a great heresy to be aga...
There's safety in free speech and a limited, Constitutional republic. Me and Thomas Jefferson against the world. Sigh. I probably shouldn't reduce you...
"If you object to stabbing six year old girls to death, you just might be a right winger." Not today. Today, he's putting people in jail who express i...
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