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- Thread starter Loren Booda
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- #51

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- #52

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- #53

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Let f:R->R be any function. No constraints except that its a function. Tell me every value of f except at x_o. Now, we take the set of all functions from R->R and define an equivalence relation where f~g if f and g differ at finitely many points. Take any member g from the equivalence class of f. Since f and g differ on a set of measure zero, f(x_o) = g(x_o) with probability 1.

Of course the fact that we can choose g relies on the axiom of choice.

:D

- #54

CRGreathouse

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Since f and g differ on a set of measure zero, f(x_o) = g(x_o) with probability 1.

I'd think that at a random point x, f(x) = g(x) with probability 1, but that wouldn't day anything about f(x_0) = g(x_0). Otherwise consider where f(x) = 1 for x = 0 and 0 otherwise, and g is uniformly 0. The probability that they agree on a random point is 1; the probability that they agree at x = 0 is 0.

This reminds me of a paper I read a few years ago, with a title along the lines of 'using the Axiom of Choice to see the future', which discussed similar techniques (as I recall!) to take a glance at a time epsilon in the future.

- #55

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- #56

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Also, any set can be well ordered by axiom of choice. And everything about the minimal uncountable (well-ordered) set.

- #57

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Let f:R->R be any function. No constraints except that its a function. Tell me every value of f except at x_o. Now, we take the set of all functions from R->R and define an equivalence relation where f~g if f and g differ at finitely many points. Take any member g from the equivalence class of f. Since f and g differ on a set of measure zero, f(x_o) = g(x_o) with probability 1.

Of course the fact that we can choose g relies on the axiom of choice.

:D

Quite interesting. I'm a little sceptical about that, but I'll think about it some more.

- #58

Hurkyl

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Why? This certainly doesn't follow from anything you said previously, because this is the first time you even mention probability. You never bother to specify what probability distribution you're using either.... (nor what the outcomes and events are)Since f and g differ on a set of measure zero, f(x_o) = g(x_o) with probability 1.

- #59

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Say you have two functions f and h that differ only at x

- #60

CRGreathouse

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Say you have two functions f and h that differ only at x_{0}; then f ~ h. Choose any member g from [f] = [h]; by your argument, f(x_{0}) = g(x_{0}) with probability 1, and h(x_{0}) = g(x_{0}) with probability 1. But these are disjoint events, so that is absurd.

Heh, I said the same thing in post #54.

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- #62

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But still there is one thing I find counterintitive more than anything I have learnt, although

belonging to physics rather than mathematics: Bernoullis fluid law, where static pressure

gets lower in more narrow passages while the speed of fluid at the same time is higher. I understand the theory, but still I find it very counterintuitive - not the least in practice, where you watch the flowing river surface

- #63

Hurkyl

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- #64

CRGreathouse

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Let f:R->R be any function. No constraints except that its a function. Tell me every value of f except at x_o. Now, we take the set of all functions from R->R and define an equivalence relation where f~g if f and g differ at finitely many points. Take any member g from the equivalence class of f. Since f and g differ on a set of measure zero, f(x_o) = g(x_o) with probability 1.

Of course the fact that we can choose g relies on the axiom of choice.

:D

This reminds me of a paper I read a few years ago, with a title along the lines of 'using the Axiom of Choice to see the future', which discussed similar techniques (as I recall!) to take a glance at a time epsilon in the future.

I found the paper I was thinking of. Citation and quote:

Christopher Hardin and Alan D. Taylor, "A Peculiar Connection Between the Axiom of Choice and Predicting the Future" (2006):

Specifically, given the values of a function on an interval (−∞, t), the strategy produces a guess for the values of the function on [t,∞), and at all but countably many t, there is an ε > 0 such that the prediction is valid on [t, t + ε). Noting that any countable set of reals has measure 0, we can restate this informally: at almost every instant t, the strategy predicts some “ε-glimpse” of the future.

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- #66

HallsofIvy

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Of course, the word "converge" is (intentionally) incorrect there.

- #67

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How about the sum of all positive integers s= 1 + 2 + 3 + 4 + ... = -1/12

- #68

Char. Limit

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How about the sum of all positive integers s= 1 + 2 + 3 + 4 + ... = -1/12

Technically that's not true, at least not in terms of a normal sum. I believe it is true if you consider the Abel sum of all positive integers.

- #69

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Technically that's not true, at least not in terms of a normal sum. I believe it is true if you consider the Abel sum of all positive integers.

It's not Abel summable, but there are several ways to derive the result such as analytic continuation of the Riemann zeta function or Ramanujan Summation, but the simplest was proposed by Euler. Perhaps more interestingly, it has been experimentally measured, in the Casimir effect for example.

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