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Do we ever distinguish between $0^+$ and $0^-$?
Incorrect proof of the infinities between 0 and 1 and 0 and 2Formal approach to (countable) prisoners and hats problem.Can we formally distinguish between actual and potential infinities?Why do we distinguish between infinite cardinalities but not between infinite values?Will a decreasing probability ever resolve favorably?Why keeping a disctinction between almost surely equal elements in probability theory?Question about an inference involving an uncountable union of null eventsDensity w.r.t. counting measure and probability mass function (discrete rv)Distinguish between gamma and log-normal distributions based on 95th percentile of a random variableWhy can’t measures be defined on uncountable powersets? Example that actually applies to probability theory
$begingroup$
I know the IEEE standard does for numerical precision reasons, but I’m asking mathematically. I ask because I’ve been thinking about how:
(VAGUE LANGUAGE BEGINS HERE) the sum of uncountably many zeros can be non-zero. For example, the length of an interval, or probability for continuous random variables, how $P(omega) = 0$ for all $omega$, but $P( 0 le omega le 1) > 0$. Perhaps it’s because the $0$ in $P(omega) = 0$ was always somehow $0^+$, something more than $0$, because it was $frac1infty$, a something divided by infinity, so increasingly closer to 0, but from the right-hand side. (VAGUE LANGUAGE ENDS HERE)
I’m aware there’s no real paradox; I haven’t defined uncountable sums, so who’s to say anything about their behaviour? However, I do wonder if there is something to the idea of tracking whether we got to $0$ through a limit that approached from the left, versus from the right (or in some other direction, or multiple directions), and use that information to make predictions about whether the ‘uncountable sum’ (again, I know I haven’t defined it) will be positive or negative.
probability-theory infinity
$endgroup$
add a comment |
$begingroup$
I know the IEEE standard does for numerical precision reasons, but I’m asking mathematically. I ask because I’ve been thinking about how:
(VAGUE LANGUAGE BEGINS HERE) the sum of uncountably many zeros can be non-zero. For example, the length of an interval, or probability for continuous random variables, how $P(omega) = 0$ for all $omega$, but $P( 0 le omega le 1) > 0$. Perhaps it’s because the $0$ in $P(omega) = 0$ was always somehow $0^+$, something more than $0$, because it was $frac1infty$, a something divided by infinity, so increasingly closer to 0, but from the right-hand side. (VAGUE LANGUAGE ENDS HERE)
I’m aware there’s no real paradox; I haven’t defined uncountable sums, so who’s to say anything about their behaviour? However, I do wonder if there is something to the idea of tracking whether we got to $0$ through a limit that approached from the left, versus from the right (or in some other direction, or multiple directions), and use that information to make predictions about whether the ‘uncountable sum’ (again, I know I haven’t defined it) will be positive or negative.
probability-theory infinity
$endgroup$
1
$begingroup$
Do you mean "the sum of uncountably many zeros can be non-zero"? That being said, have a look at non-standard analysis.
$endgroup$
– Arthur
Mar 13 at 10:20
$begingroup$
We distinguish between $lim_xto0^+f(x)$ and $lim_xto0^-f(x)$. The rest, I think, is gibberish.
$endgroup$
– Gerry Myerson
Mar 13 at 11:51
add a comment |
$begingroup$
I know the IEEE standard does for numerical precision reasons, but I’m asking mathematically. I ask because I’ve been thinking about how:
(VAGUE LANGUAGE BEGINS HERE) the sum of uncountably many zeros can be non-zero. For example, the length of an interval, or probability for continuous random variables, how $P(omega) = 0$ for all $omega$, but $P( 0 le omega le 1) > 0$. Perhaps it’s because the $0$ in $P(omega) = 0$ was always somehow $0^+$, something more than $0$, because it was $frac1infty$, a something divided by infinity, so increasingly closer to 0, but from the right-hand side. (VAGUE LANGUAGE ENDS HERE)
I’m aware there’s no real paradox; I haven’t defined uncountable sums, so who’s to say anything about their behaviour? However, I do wonder if there is something to the idea of tracking whether we got to $0$ through a limit that approached from the left, versus from the right (or in some other direction, or multiple directions), and use that information to make predictions about whether the ‘uncountable sum’ (again, I know I haven’t defined it) will be positive or negative.
probability-theory infinity
$endgroup$
I know the IEEE standard does for numerical precision reasons, but I’m asking mathematically. I ask because I’ve been thinking about how:
(VAGUE LANGUAGE BEGINS HERE) the sum of uncountably many zeros can be non-zero. For example, the length of an interval, or probability for continuous random variables, how $P(omega) = 0$ for all $omega$, but $P( 0 le omega le 1) > 0$. Perhaps it’s because the $0$ in $P(omega) = 0$ was always somehow $0^+$, something more than $0$, because it was $frac1infty$, a something divided by infinity, so increasingly closer to 0, but from the right-hand side. (VAGUE LANGUAGE ENDS HERE)
I’m aware there’s no real paradox; I haven’t defined uncountable sums, so who’s to say anything about their behaviour? However, I do wonder if there is something to the idea of tracking whether we got to $0$ through a limit that approached from the left, versus from the right (or in some other direction, or multiple directions), and use that information to make predictions about whether the ‘uncountable sum’ (again, I know I haven’t defined it) will be positive or negative.
probability-theory infinity
probability-theory infinity
edited Mar 13 at 10:24
Yatharth Agarwal
asked Mar 13 at 10:16
Yatharth AgarwalYatharth Agarwal
542418
542418
1
$begingroup$
Do you mean "the sum of uncountably many zeros can be non-zero"? That being said, have a look at non-standard analysis.
$endgroup$
– Arthur
Mar 13 at 10:20
$begingroup$
We distinguish between $lim_xto0^+f(x)$ and $lim_xto0^-f(x)$. The rest, I think, is gibberish.
$endgroup$
– Gerry Myerson
Mar 13 at 11:51
add a comment |
1
$begingroup$
Do you mean "the sum of uncountably many zeros can be non-zero"? That being said, have a look at non-standard analysis.
$endgroup$
– Arthur
Mar 13 at 10:20
$begingroup$
We distinguish between $lim_xto0^+f(x)$ and $lim_xto0^-f(x)$. The rest, I think, is gibberish.
$endgroup$
– Gerry Myerson
Mar 13 at 11:51
1
1
$begingroup$
Do you mean "the sum of uncountably many zeros can be non-zero"? That being said, have a look at non-standard analysis.
$endgroup$
– Arthur
Mar 13 at 10:20
$begingroup$
Do you mean "the sum of uncountably many zeros can be non-zero"? That being said, have a look at non-standard analysis.
$endgroup$
– Arthur
Mar 13 at 10:20
$begingroup$
We distinguish between $lim_xto0^+f(x)$ and $lim_xto0^-f(x)$. The rest, I think, is gibberish.
$endgroup$
– Gerry Myerson
Mar 13 at 11:51
$begingroup$
We distinguish between $lim_xto0^+f(x)$ and $lim_xto0^-f(x)$. The rest, I think, is gibberish.
$endgroup$
– Gerry Myerson
Mar 13 at 11:51
add a comment |
0
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1
$begingroup$
Do you mean "the sum of uncountably many zeros can be non-zero"? That being said, have a look at non-standard analysis.
$endgroup$
– Arthur
Mar 13 at 10:20
$begingroup$
We distinguish between $lim_xto0^+f(x)$ and $lim_xto0^-f(x)$. The rest, I think, is gibberish.
$endgroup$
– Gerry Myerson
Mar 13 at 11:51