Proof of Immortality III

[jsfisher]

For a real number r, r/∞ is not defined.

And that was Loss Leader's point, too. So we all agree.

However, r/n approaches zero as n approaches infinity, so the expression has a value in the limit.

The limit of R/N as N approaches infinity has a value. R/infinity (without further qualification) does not.

That's how R -- a popular statistical modeling language -- is able to confidently evaluate it.

R is confident? Curious that. Nonetheless, R produces a result in accordance with what its creators decided was useful behavior for the class of problems R was designed to address.

R is not, however, the authority for mathematical meaning as was alleged.

There's a nuance between whether it's defined and whether it can be said to have a value. Values derived "in the limit" are still valid for many purposes, otherwise calculus wouldn't work.

And they can be invalid for many purposes. Consider a die of infinitely many sides, one side for each of the positive integers. What is the probability a roll will yield a 5? Zero. What is the probability it will yield a number of 5 or less? Zero, again, derived from the sum of the individual probabilities = 0+0+0+0+0.

What is the probability a roll yields a number, some number, any number at all? One, of course, yet the sum of the individual probabilities, 0+0+0+..., is zero. Limits have failed us.

If I am recalling correctly, I think this is the point Loss Leader was trying to explore. Jabba was asserting a high probability of his existence under the ~H assumption. However, ~H may include infinitely many possible realities (and not necessarily countable many realities, either), and so Jabba's hand-waving probability estimates are more than suspect.

 

[caveman1917]

Well, you can't divide by infinity. You can't ever divide infinity by infinity. What misstake am i making?

A probability function is a function from a sigma-algebra to the unit interval. The probability of an event can not have two values. There are ways in which to assign probabilities to events in an infinite set of outcomes. It will depend on the specifics of the set of outcomes. If, for example, the set of outcomes is countably infinite then a probability function could be: P(X = n) = 2^{-n}

 

[caveman1917]

There are plenty of extensions of the real numbers in which infinity and operations on it are defined. The problem with Loss Leader's claim wasn't so much the infinity thing but that he claimed the probability of an event to have multiple values.

 

[JayUtah]

And that was Loss Leader's point, too. So we all agree.

Yep. It's just important to reconcile the notions of "is defined" and "has a value," which I think we've collectively done. Well, at least to my satisfaction. It's important because the propriety of the argument, in my opinion, varies according to that distinction.

Nonetheless, R produces a result in accordance with what its creators decided was useful behavior for the class of problems R was designed to address.

Statistical analysis is the class of problems R was designed to address. Jabba is attempting statistical analysis. When asked to evaluate the expression 1/∞, it can do so -- not because it is defined but because a single practical value for the expression (without larger context) can be had for all real denominators. Give it ∞/∞, and the story is different.

R is not, however, the authority for mathematical meaning as was alleged.

I'm not sure it was alleged in so many words, but rest assured I stand on your side. R is not an authority in resolving nuances in mathematical conceptualization. Being able to evaluate an expression practically is not the same as asserting an expression has a defined outcome by the rules of the relevant operators. That's one of the problems limits were formulated to solve. As you illustrate in a sample problem, the model still has to be constructed properly so that the concept of infinity is correctly handled. Just because R or some other system can evaluate it by assuming a limit-theory approach doesn't mean the model properly corresponds to reality or that the theory was appropriate to that case.

And they can be invalid for many purposes.

Indeed, and for the record I consider the purpose to which Jabba is putting it invalid. He's trying to substitute ∞ for "a very large number." There is one error in assuming the value in question must be very large, and an entirely separate error in the substitution.

If I am recalling correctly, I think this is the point Loss Leader was trying to explore.

That's my recollection too, which in my mind requires the explicit distinction between the conceptual understanding of infinity as the cardinality of the natural numbers and the arithmetic properties we can apply to it for some practical purposes. As you're well aware, Jabba loves to equivocate. Hopefully through this chapter of the discussion we can prevent him from doing so.

Jabba's hand-waving probability estimates are more than suspect.

Everyone agrees with that.

 

[Mojo]

If I am recalling correctly, I think this is the point Loss Leader was trying to explore. Jabba was asserting a high probability of his existence under the ~H assumption. However, ~H may include infinitely many possible realities (and not necessarily countable many realities, either), and so Jabba's hand-waving probability estimates are more than suspect.

One of the fatal flaws in Jabba's argument is that he has defined H, not as the hypothesis that he has a single finite lifetime, but as the hypothesis that his existence is the result of chance and he has a single finite lifetime.

As he himself has defined it, ~H includes hypotheses under which Jabba only has a single finite lifetime.

 

[caveman1917]

And they can be invalid for many purposes. Consider a die of infinitely many sides, one side for each of the positive integers. What is the probability a roll will yield a 5? Zero. What is the probability it will yield a number of 5 or less? Zero, again, derived from the sum of the individual probabilities = 0+0+0+0+0.

What is the probability a roll yields a number, some number, any number at all? One, of course, yet the sum of the individual probabilities, 0+0+0+..., is zero. Limits have failed us.

Whatever it is you're talking about that is zero for each outcome, it ain't a probability.

 

[caveman1917]

There is also an important difference between outcomes and events. Outcomes don't have probabilities, events do. As long as we stay with an at-most countably infinite set of outcomes we can construct a sigma-algebra (set of events) where each outcome is its own event. When we have an uncountable set of outcomes this is not possible. It's not technically correct to say that outcomes in such a set have a probability of zero, rather they have no probability.

 

[jt512]

IEverything I've learned about the set of real numbers and about division is now cast in doubt. I feel so lost. Peano, you have betrayed me.

I'm always happy to help.

For a real number r, r/∞ is not defined.

Yeah, actually it is, at least in calculus. For any real number r, r / ∞ = 0. See, for instance, Definition 1.23(a) in Rudin's Principles of Real Analysis.

 

[jt512]

R is not, however, the authority for mathematical meaning as was alleged.

Check your sense of humor. No one "alleged" that R was an authority on division by ∞. For an authority, see Rudin's Principles of Mathematical Analysis.

Consider a die of infinitely many sides, one side for each of the positive integers. What is the probability a roll will yield a 5? Zero. What is the probability it will yield a number of 5 or less? Zero, again, derived from the sum of the individual probabilities = 0+0+0+0+0.

Some of those countable sides had better have positive probability, and the sum of the probabilities of all the sides had better be 1. If not, probabilities of the sides are undefined.

What is the probability a roll yields a number, some number, any number at all? One, of course, yet the sum of the individual probabilities, 0+0+0+..., is zero. Limits have failed us.

Your understanding of probability has failed you. You need to forget the Peano axioms and look up the Kolmogorov ones.

 

[caveman1917]

Yeah, actually it is, at least in calculus. For any real number r, r / ∞ = 0. See, for instance, Definition 1.23(a) in Rudin's Principles of Real Analysis.

You are talking about the extended real numbers, not the real numbers. Let R be the set of real numbers and *R be the set of extended real numbers. Let alpha be a map R -> *R which associates every real number in R with its counterpart in *R. Then

For any r in R: r / inf is undefined
For any r in *R such that r in Img(alpha): r / inf = 0

I've checked your reference and noticed that a clear distinction isn't being made between a real number and its associated extended real number, which might be the cause of the confusion here.

ETA: the extended real numbers do not include the real numbers, they include numbers which are associated with real numbers, the sets of real numbers and extended real numbers are distinct.

 

[jt512]

You are talking about the extended real numbers, not the real numbers.

I know (again).

Let R be the set of real numbers and *R be the set of extended real numbers. Let alpha be a map R -> *R which associates every real number in R with its counterpart in *R. Then

For any r in R: r / inf is undefined

As stated, that is false. R ⊂ R*, so for any r ∈R r / ∞ = 0.

I think what you are trying to say is

In R r / ∞ is undefined.

But that is not what you said.

I've checked your reference and noticed that a clear distinction isn't being made between a real number and its associated extended real number, which might be the cause of the confusion here.

Rudin is quite clear about that (and everything else, for that matter).

ETA: the extended real numbers do not include the real numbers, they include numbers which are associated with real numbers, the sets of real numbers and extended real numbers are distinct.

I don't know where you could have gotten that idea. The extended number system is defined as R ∪ {–∞, +∞} (together with an ordering property).

 

[jt512]

There is also an important difference between outcomes and events. Outcomes don't have probabilities, events do. As long as we stay with an at-most countably infinite set of outcomes we can construct a sigma-algebra (set of events) where each outcome is its own event. When we have an uncountable set of outcomes this is not possible.

Of course it is possible. Let the sample space be the real line. Then the σ-algebra that includes the sets (a, b), (a, b], [a, b], and [a, b) for all real numbers a and b includes the set {b} for every real number b.

It's not technically correct to say that outcomes in such a set have a probability of zero, rather they have no probability.

Given the σ-algebra described above, for all b in R P({b}) = 0.

 

[jsfisher]

One of the fatal flaws in Jabba's argument is that he has defined H, not as the hypothesis that he has a single finite lifetime, but as the hypothesis that his existence is the result of chance and he has a single finite lifetime.

As he himself has defined it, ~H includes hypotheses under which Jabba only has a single finite lifetime.

He also pulls a bait-and-switch on the whole existence thing. The probability of his existence under H is based on the likelihood the specific set of subatomic particles came into alignment at a specific place and time to yield the particular human we know as Jabba.

Under the alternate hypothesis, Jabba changes view to use his soul (or whatever) for his existence, not the meat sack.

This is why Jabba cannot/will not address the contradiction in his numbers where he asserts P(meat sack) < P(meat sack + soul), because he is really thinking in terms of P(meat sack) < P(soul). He simply does not see that he shifted meaning.

 

[caveman1917]

As stated, that is false. R ⊂ R*, so for any r ∈R r / ∞ = 0.

I think what you are trying to say is

In R r / ∞ is undefined.

But that is not what you said.
{...}
I don't know where you could have gotten that idea. The extended number system is defined as R ∪ {–∞, +∞} (together with an ordering property).

Then the problem is the double use of the "/" symbol for both /_R and /_R* (ie division as defined over R or over R*). Either way, something has got to give here, as "for any r ∈R r / ∞ = 0" can not be correct as stated - I could after all just define another extension of R in which that doesn't hold.

In this case:
for all r in R: r /_R inf is undefined
for all r in R: r /_R* inf = 0

 

[caveman1917]

Of course it is possible. Let the sample space be the real line. Then the σ-algebra that includes the sets (a, b), (a, b], [a, b], and [a, b) for all real numbers a and b includes the set {b} for every real number b.

Given the σ-algebra described above, for all b in R P({b}) = 0.

Yes that is true, don't know what I was thinking there. Do you even need the half-open and closed intervals? Seems you could just define it on the open intervals (or the closed ones for that matter).

 

[jsfisher]

Check your sense of humor. No one "alleged" that R was an authority on division by ∞.

Actually, you did. In this post. If you meant the reference to R to be a funny, a smilie or two may have helped others understand that.

For an authority, see Rudin's Principles of Mathematical Analysis.

As caveman1917 has pointed out, this is not for the reals and normal arithmetic.

Some of those countable sides had better have positive probability, and the sum of the probabilities of all the sides had better be 1. If not, probabilities of the sides are undefined.

Really? So for a fair die with countably infinitely many sides, what would be the probability of any particular side coming up?

Switch to uncountably infinite if you prefer. For some continuous random variable that is uniformly distributed between 0 and 10, what is the probability its value is 4.318?

 

[caveman1917]

Really? So for a fair die with countably infinitely many sides, what would be the probability of any particular side coming up?

Yes, really. If P = 0 for all n then P is not a probability function, calling it a "probability" would be incorrect. Or differently, there is no such thing as a "fair die with countably infinitely many sides".

Switch to uncountably infinite if you prefer. For some continuous random variable that is uniformly distributed between 0 and 10, what is the probability its value is 4.318?

It's something else when you switch from countably infinite to uncountably infinite, the sum becomes an integral. To stick with your example, you'd have int_0^10 P = 0 in which case P wouldn't be a probability function either.

This is what I was trying to say earlier but badly messed it up, as jt512 pointed out, with my "no probability for singleton events in an uncountable sample space".

 

[jt512]

Actually, you did. In this post. If you meant the reference to R to be a funny, a smilie or two may have helped others understand that.

I thought the wording made it reasonably clear that it was not meant literally.

As caveman1917 has pointed out, this is not for the reals and normal arithmetic.

Dividing by ∞ implies you're in the extended real number system. The reals are a subset of the extended reals, so any real number can be divided by ∞, despite what Caveman thinks.

Really? So for a fair die with countably infinitely many sides, what would be the probability of any particular side coming up?

You tell me. It's your die. Define a probability function that satisfies the Kolmogorov axioms over its sample space. The axioms imply that the probabilities p_i of the sides i, i=1,2,..., must sum to 1. Clearly, then, the p_i cannot all be equal.

Switch to uncountably infinite if you prefer. For some continuous random variable that is uniformly distributed between 0 and 10, what is the probability its value is 4.318?

P(X=4.318 (exactly)) = 0.

 

[caveman1917]

Dividing by ∞ implies you're in the extended real number system.

Claim: for all r in R_0: r / 0 = ∞

What would your response be?

 

[jt512]

Yes that is true, don't know what I was thinking there. Do you even need the half-open and closed intervals? Seems you could just define it on the open intervals (or the closed ones for that matter).

Yeah, supposedly, you only need one type of interval: open, closed, or half-open, and all the other sets follow from the properties of a σ-algebra, but I'm not sure I could figure out how to get them all.