Limiting ourselves for the sake of convenience to just Achilles and the tortoise, what exactly is there to debate about? — tim wood
So there is no need to involve an object named 'inf'. It's a more parsimonious approach. — TonesInDeepFreeze
I don't see a need for disagreement here. — TonesInDeepFreeze
You can notate as you wish; and I can say why I also use that notation but like to point out that 'inf' is dispensable when we unpack, which has a pedagogical purpose: Many people, such as in Internet threads, are clueless about axiomatic, rigorous mathematics, so they stubbornly assert that the notation with 'inf' must mean that it names a certain entity. So it is instructive to explain that it is merely contextual notation that does not invoke any entities other than real numbers and a function. — TonesInDeepFreeze
Can we calculate the contribution of the w-th term to the series? — MoK
Only reason can understand the infinite, not imagination (it would seem). — Gregory
Not much, I should think. I wonder what odds Zeno might give.What should be the payoff if you bet 1000 euros on Achilles. — TonesInDeepFreeze
Thanks for the clarification. Can we calculate the contribution of the ω
�
th term to the series? — MoK
I use reason to formally conceive that there are infinite sets. It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle). — TonesInDeepFreeze
It's easy: I understand the property of being a natural number; and I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle). — TonesInDeepFreeze
One problem though does haunt me: Every arithmetical statement is either true or false. — TonesInDeepFreeze
Reason is utterly insufficient to determine if there are infinite sets.
The only thing that guarantees the existence of an infinite set is the axiom of infinity. — fishfry
If we adopt the axiom of infinity, there are infinite sets. If we reject the axiom of infinity, there are no infinite sets. It's as simple as that. — fishfry
The existence of an infinite set is purely a matter of accepting or rejecting the assumption that says there is an infinite set. — fishfry
Your principle leads directly to a contradiction. The restriction needed to patch the problem is restricted comprehension, which would already require there to be an infinite set that's a superset of the infinite set you wish to conjure. — fishfry
One problem though does haunt me: Every arithmetical statement is either true or false.
— TonesInDeepFreeze
No. Just no. Given an arithmetical statement (a syntactic entity) AND AN INTERPRETATION of the symbols of that statement (semantics) then the statement becomes true or false. — fishfry
I should have counted on someone not recognizing that my remarks are merely in the spirit of personal reflection. — TonesInDeepFreeze
"When caught in a material error, I just claim I didn't really mean it that way." — fishfry
Too tedious to likewise mock and debunk every single point you made in this post. I stand by my previous remarks. — fishfry
(1) Rejecting the axiom of infinity does not entail that there are no infinite sets. Rather, both rejecting the axiom of infinity and adopting the negation of the axiom of infinity entails that there are no infinite sets. (This pertains to ZF.) — TonesInDeepFreeze
I don't know the relevance you intend with that quote. — TonesInDeepFreeze
You are the one who attacks every little technical inaccuracy anyone makes, even if their overall meaning is obvious. But you don't hold yourself to that same standard. — fishfry
even if their overall meaning is obvious — fishfry
but also hope that some reasonable liberty is granted — TonesInDeepFreeze
Was it not perfectly clear the other day that the usual order on the integers carries over to the extended integers — fishfry
But today you were flat out wrong to claim that "reason" shows that there are infinite sets. — fishfry
Likewise your invocation of unrestricted comprehension to justify that falsehood. — fishfry
I couldn't find any reasonable liberty there so I corrected your material errors. — fishfry
It's a full time job, let me tell you. — fishfry
I don't get it. Or maybe adducing that quote is just your way of saying "yeah yeah" ironically. If so, whatever. — TonesInDeepFreeze
Whoa. I did not at all invoke unrestricted comprehension. — TonesInDeepFreeze
I understand the idea that given a property, there is the set of things that have that property (with some restrictions on that principle). — TonesInDeepFreeze
At this point I'm just trolling you. — fishfry
the required restriction would negate your point. — fishfry
Without the axiom of infinity, THERE IS NO SET containing all the natural numbers. — fishfry
"0 = the limit of the function 1/x where x ranges over the positive natural numbers," what do you mean by limit? Do you mean that for all epsilon there exists a delta such that ... etc? Where does your definitional parsimony end? If I say, "Let X be a topological space," should I replace that with, "Let X be a set along with a collection of its subsets satisfying ..." and then repeat the definition of a topology? — fishfry
'lim' is a variable binding operator, but it can be reduced to a regular operation symbol:
Df. If g is a function from the set of positive natural numbers into the set of real numbers, and there exists a unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y, then Lg = the unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y
You see that 'L' is an operation symbol that takes only finitely many arguments - in this case, one argument.
The argument itself is an infinite set (an infinite sequence in this case), which is okay, because the operation symbol takes only finitely many arguments - in this case, one argument.
And we prove, regarding the function f we previously defined:
Lf = 0
In everyday parlance, "the limit of f is 0" [...] — TonesInDeepFreeze
Given that I already have a definition of 'the limit', I can just say 'the limit' without reciting again its definiens. — TonesInDeepFreeze
This is what I'm confused about with your objection to the extended reals (or integers, etc.) — fishfry
I noted that the extended reals are essentially a notational convenience, and we could live without them. — fishfry
And you seemed to be arguing that because we COULD live without them, then we SHOULD live without them. — fishfry
The definition of limit is rather involved, at least for people first encountering it, involving epsilon and delta and universal and existential quantifiers and so forth. — fishfry
By your logic (as I understand it), it would be parsimonious (a virtue, I gather, but I'm not sure why) to dispense with it, and do the raw epsilonics every time we want to mention a limit. — fishfry
You object to the use of the extended reals as a notational convenience for infinite limits and limits at infinity — fishfry
I don't understand your point about the extended reals. — fishfry
parsimonious — fishfry
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