3 ms·
Yes, the computable, countbale universe is incompatible with ZFC. But ZFC is math, not computation or physics. ZFC is consistent with itself, which is good eno
by lupire 4y ago
Yes, the computable, countbale universe is incompatible with ZFC.
But ZFC is math, not computation or physics. ZFC is consistent with itself, which is good enough for math.
You don't even need AoC to run into this. Even the "tame" smooth functions of classical physics (like f(x)=x^2) are a non-physical approximation to discrete reality.
- red_trumpet 4y agoBold to assume that reality is discrete.
- lupire 4y agoEven if it's not discrete at the bottom of QFT or something, it is discrete at the level where physics is applied in classical mechanics.
- posterboy 4y agoBold to assume that "it" is reality. Conversely, classical mechanics is discrete at the level where physics is applied. That it is applied to realia goes without saying. But it is not applied to reality as a whole.
- Tainnor 4y agoWell, it's bold to assume anything about reality, because we only ever experience individual aspects of it. We can only try to model what we can observe. But as a counter to the assertion that "AC must be wrong because we can't really experience Banach-Tarski in the real world", it's perfectly valid to comment that "well, we also can't really observe continuous functions R -> R either". Or have you ever seen a least upper bound in real life?
- syzarian 4y agoEverything appears to be discrete except possibly space-time. Is there a reason to believe space-time isn’t discrete other than using manifolds makes things easier mathematically?
- leni536 4y agoThere is no reason to believe that spacetime is anything but an approximation, which breaks down at Planck-length scales. We have events, and causal relationship between events. Events may not have precise spacetime coordinates.
- lanstin 4y agoI believe the numbers in QFT are continuous. The electrons and what not are quantized but quantized over complex number spaces. U less one be Unless the fine structure constant or speed of light, or other apparent constants of the theories, are computable some how? Disclaimer: I can only get a few chapters into a QFT book without it all becoming meaningless to me.)
- syzarian 4y agoI believe the universe is finite and so there can only be a finite number of realizations of quantum numbers. I have no real intuition in physics or training in physics so I know I could be completely wrong. It seems to me though that our experiences are such that believing the universe is discrete in all things is not completely unreasonable. Though this belief of mine might itself be unreasonable.
- andrewla 4y agoThe mathematical notion of "discrete" does not require that there be big gaps, or that there be fundamental limits to precision. The rationals are discrete (i.e. they are countable and have measure zero) but they are "continuous" in every way that matters -- between any two rationals there is another rational. Even the computable/constructible/rationally-approximable numbers are discrete, and those sets contain basically all the numbers that are possible to compute (by definition) but they are still "discrete", forming a set of measure zero, The idea of the continuum is the unintuitive one; the idea that there are numbers that cannot be constructed or computed is just kind of silly
- andrewla 4y agoIf by "discrete" you mean rejecting the continuum, then I don't think it's as scary as it sounds. The rationals do not form a continuum, but they are "effectively" continuous -- between any two of them there is another. There are no "leaps" between them, and no concept of consecutive numbers. Computable numbers are similarly countable but allow the expression of numbers that can only be approximated by rationals. It's the reals that require "bold" assumptions -- numbers exist which are not possible to compute even approximately?
- Tainnor 4y ago> ZFC is consistent with itself, which is good enough for math. That's what we believe, but we don't know it. Nor do we have any hope of proving it in some reasonable way (via Gödel's second incompleteness theorem).
- TuringTest 4y agoMath is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. Math is but a language to express your thoughts in a very precise and rational way. When you find out that two of your carefully expressed thoughts contradict each other, you tweak one or both so that they are no longer contradictory. That's why "the barber who shaves those who don't shave themselves" and "the set of all sets that don't contain themselves" are not considered valid sets anymore.
- Tainnor 4y ago> Math is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. The former doesn't follow from the latter, because we may simply not yet have found any inconsistency, even though it exists. edit: Not sure if your point is just "if someone finds a contradiction in ZFC, it doesn't break maths, we'll just find a different foundation". Because I agree. But GP's point was about ZFC, a concrete set of axioms, being consistent.