But then these numbers - growth constants like e and phi – are ratios and so are dimensionless unit 1 values more than they are some weird real number. — apokrisis
Adding to my response about the particular paradoxes. Even if we granted that they indicate flaws in the concept of infinitude, then that is a concept of infinitude extended beyond set theory into imaginary states of affairs for which set theory should not take blame. Those paradoxes don't impugn set theory itself. — TonesInDeepFreeze
I know so little of cosmology that I don't know how to dispel my bafflement that the universe could be finite or my bafflement that the universe could be infinite. — TonesInDeepFreeze
If there is any mathematical reasoning that can be considered safe, then it's manipulation of finite strings of objects or symbols (whether concrete sticks on the ground, or abstract tokens)...if I really had to, I could fall back to extreme formalism by taking the theorem to be utterly uninterpreted, but a formula nevertheless to be used in mathematical reckoning. — TonesInDeepFreeze
as actual infinities (rather infinitesimals) were banished from (mainstream) calculus and replaced with potential infinities (through limits) — keystone
I think it's quite easy to imagine a closed finite universe, for example a sphere of finite radius. It's a lot harder to fit in one's mind a sphere of infinite radius. — keystone
by and large I am completely convinced that the vast majority of modern math would retain it's value even IF actual infinities were banished. — keystone
What was the 19th century analysis resolution to Zeno's paradox?
— keystone
Infinite summation: convergence of an infinite sequence to a limit. — TonesInDeepFreeze
Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms?
— keystone
I don't conceive of an infinite set of physical rooms.
As to sets, I already mentioned that I am not bothered that the squares (a proper subset of the naturals) is 1-1 with the naturals. — TonesInDeepFreeze
I don't think calculus needs actual infinity to work.
— keystone
It does in its common form. — TonesInDeepFreeze
I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either.
— keystone
It wouldn't be ZFC then. — TonesInDeepFreeze
The imaginary hotelier can do that also. — TonesInDeepFreeze
potential vs actual infinity — apokrisis
If we want for it to be provable that there does not exist an infinite set, then we need axioms to do that. — TonesInDeepFreeze
I believe that irrationals are algorithms which describe this mysterious other object - continua.
— keystone
I've already raised this point, asking if sqrt(2) is the exception or the rule. Higher dimensional generators could produce generators of some number that looks to be an irrational point of the line. But then these numbers - growth constants like e and phi – are ratios and so are dimensionless unit 1 values more than they are some weird real number.
The status of any regular irrational seems different. They would lack generators apart from decimal expansion. Something else is going on. — apokrisis
I don't believe there is a fundamental length since any length can be divided.
— keystone
Can the Planck length be divided? Not without curling up into a black hole.
What you believe and what the Universe would like to tell you seem two different things. Who wins? — apokrisis
I can imagine a mind that lives in a 4D universe that can picture a 4D hypersphere as easily as a sphere.
— keystone
Yea, nah. I'm not buying these feats of your imagination. — apokrisis
the one where the world is "thinking itself" into definite being in ontic structural fashion. — apokrisis
The hidden rabbit or seagull is merely hidden while the brain finds a way to suppress the shapelessness of the coloured pixels from the intelligibility of a depth perception-based contour. — apokrisis
The page contains the potential of infinite images,
— keystone
The problem is that it doesn't. It plays on a dichotomous rivalry of brain subsystems. You have to switch off the one and employ the other. The search is for the single hidden interpretation. Only one of the two points of view can spot it. — apokrisis
This is why the brain is not a computer. — apokrisis
Yes, this will seem very counterintuitive. The simplest way I can explain it in a non-technical fashion is that selecting any non-zero probability for each number will force us to add up way over 100%, because there are infinitely many other "participants" (numbers), which means the only probability we can assign to each participant is zero. — Kuro
There's actually a way out of this being nonstandard analysis — Kuro
(1) 'finitism' has different senses. — TonesInDeepFreeze
(2) Perhaps it is not necessary to have infinite sets for an axiomatization of mathematics for the sciences. It's just that in order to evaluate a non-infinitistc axiomatization, we need to have it specified. — TonesInDeepFreeze
I can draw a line with open ends on a piece of paper and label the ends negative and positive infinity. This unbounded object is entirely finite.
— keystone
You can draw a sign that you then interpret in a certain habitual fashion. The issue then is how does this sign relate you to the reality beyond. Does is create a secure bridge? Or is it wildly misleading? — apokrisis
If maths has been left behind in this grand and still unfolding adventure, tough shit. — apokrisis
In set theory, there is no completeness axiom. Rather, we prove as a theorem that the system of reals is a complete ordered field. — TonesInDeepFreeze
And yes one might want for the axioms to be intuitively correct ("true") even if the theorems might be surprising. And with set theory, people's mileages vary. I find the axioms of set theory to be exemplary in sticking to only principles that are in concordance with the intuitive notion of 'sets'. — TonesInDeepFreeze
Does finitism mean some domains of math vanish into thin air?
Undefinable real numbers have no place in my view. — keystone
My point is the rule not the exception. — keystone
It has not been determined whether space is discrete or continuous (LINK). I'm inclined to believe that the planck length is a limitation that is applied to measurement, not the divisibility of space itself. — keystone
It sounds like you're saying that the rabbit exists even when not observed. — keystone
If I handed you a blank white piece of paper you could argue that it is a picture of a polar bear playing in the snow but I would argue that it only contains the potential to be such a picture, and it would actually be that picture only once you cut out the bear figure with scissors. — keystone
Perhaps I'm missing your point. Do you agree that given a continuum there's infinite potential to how you cut it up? — keystone
I agree with Max Tegmark when he said 'we should reject carbon-chauvinism'. — keystone
Must an ordered field necessarily be a field of numbers? — keystone
Could it instead be a field of equations? — keystone
the axioms of set theory are not in concordance with the intuitive notions of 'finite sets' — keystone
And since the only sets we ever work with directly are finite, I think we should be cautious accepting axioms that oppose them. — keystone
Instead of saying 'there exists an infinite set' I would be comfortable saying 'there exists an algorithm that describes an infinite set'. — keystone
My argument is that the whole potential vs actual infinity thing comes from the fact that our ideas about numbers are based on systems of constraints. — apokrisis
No, limits use infinite sets. The standard axiomatization of analysis is ZFC. Ordinary modern analysis is decidedly infinitisitic. Maybe you're thinking of the banishment of infinitesimals? — TonesInDeepFreeze
an infinite set and its proper subset have the same cardinality — Agent Smith
in colloquial terms means a part is equal to the whole — Agent Smith
there are peeps who say this is exactly what the mathematics says — Agent Smith
∞ maybe the internal combustion engine of math - creates more problems than solutions. — Agent Smith
You believe that math itself has some fundamental limits, perhaps a frequency, a duration, or a length. — keystone
Every computer has its limits. It has a finite memory so there is a limit to the size of the numbers that it can store. — keystone
Mathematical objects don't exist eternally in the Platonic realm. They exist when we (computers) compute! — keystone
A sphere has infinitely many points in it.
And is there such a thing as a sphere with an infinite radius? If I'm not mistaken the radius of a sphere is a real number, right? — TonesInDeepFreeze
the informal definitions that us engineers were taught didn't use sets. — keystone
a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value — keystone
I can think of x^2+y^2=1 without having to think of any points — keystone
The points come from the circle, not the other way around. — keystone
No, there is no such thing as a sphere with infinite radius...that's explains why I can't imagine it. — keystone
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