What's The Chance You'll Answer This Poll

[Ocelot]

The problem with your statement is that it's still true if you replace "unlikely" with "likely" because you are talking about the fact that your assessment of the situation might be flawed. Taking that into account, the original thing being discussed is still correct:

"something that is very likely is exactly as possible as something that is very unlikely."

Yes Clingers example of Hilbert's tenth problem exemplifies that.

 

[quarky]

[latex]
\[ \frac{\pi(x)}{x-\pi(x)} \sim \frac{1}{(\log x)-1} \]
[/latex]


Can't help you with that one.

Isn't 1/1 another way of seeing it? Both being infinite?

 

[W.D.Clinger]

[latex]
\[ \frac{\pi(x)}{x-\pi(x)} \sim \frac{1}{(\log x)-1} \]
[/latex]

Isn't 1/1 another way of seeing it? Both being infinite?

When the real numbers are extended with plus and minus infinity, as in the IEEE-754 or IEEE-754-2008 standards for floating point arithmetic, the result of dividing an infinity by an infinity is not mathematically well-defined. That's why both of those IEEE standards add special NaN values, where NaN means "not a number". So NaN would be a better answer than 1/1.

The formula I gave you is more useful because it tells you the approximate ratio of primes to non-primes in the range from 0 to x.

If you take the limit of that formula as x increases without bound ("goes to infinity" in the misleading vernacular), then you'll find that the ratio of primes to non-primes becomes arbitrarily close ("converges") to zero. Hence zero would be an even better answer than NaN or 1/1, if you're asking about the ratio taken over all natural numbers.

So I could have answered your question by writing "0", but I thought the formula I gave would be more informative, and it already implies 0 in the limit as x increases toward infinity.

[size=-1](Although this post may violate Rule 11, the moderators might allow it to stand long enough for you to read it, or they might split our posts into a new thread.)[/size]​

 

[quarky]

When the real numbers are extended with plus and minus infinity, as in the IEEE-754 or IEEE-754-2008 standards for floating point arithmetic, the result of dividing an infinity by an infinity is not mathematically well-defined. That's why both of those IEEE standards add special NaN values, where NaN means "not a number". So NaN would be a better answer than 1/1.

The formula I gave you is more useful because it tells you the approximate ratio of primes to non-primes in the range from 0 to x.

If you take the limit of that formula as x increases without bound ("goes to infinity" in the misleading vernacular), then you'll find that the ratio of primes to non-primes becomes arbitrarily close ("converges") to zero. Hence zero would be an even better answer than NaN or 1/1, if you're asking about the ratio taken over all natural numbers.

So I could have answered your question by writing "0", but I thought the formula I gave would be more informative, and it already implies 0 in the limit as x increases toward infinity.

[size=-1](Although this post may violate Rule 11, the moderators might allow it to stand long enough for you to read it, or they might split our posts into a new thread.)[/size]​

I appreciate your explanation. Truly.
No mod could ever invoke rule 11 in this case, could they?
Would they even know what we're discussing?