post natal depression.

Originally posted by lucifershammer


Simple question, Doctor - is it possible or impossible to pick 0.5?

Since you're a fan of straight answers, perhaps you can put your money where your mouth is and give me one.

Without this qualification: "If you can pick any real number between 0 and 1"

it is impossible.


With that qualification, it is possible merely by supposition.

Originally posted by DoctorScribbles
I said that, after saying that the very formulation of the question is inconsistent.

Your premise ("If you can pick..." ) asserts that picking a real number from a set of them is possible. It's not. For any experiment in which you think you have, you have actually conducted a different experiment than the one described.

Suppose you try to do ...[text shortened]... o discuss the real world, the one in which Adam sinned with probability greater than 0.

That's just hooey. The formulation of the question is a straightforward application of the axiom of choice (AC). If you want to reject AC because you want to maintain your axiom on the equivalence of possibility and probability, that's your decision. I don't draw the latter equivalence and I see no reason to reject AC.

Originally posted by lucifershammer
That's just hooey. The formulation of the question is a straightforward application of the axiom of choice (AC). If you want to reject AC because you want to maintain your axiom on the equivalence of possibility and probability, that's your decision. I don't draw the latter equivalence and I see no reason to reject AC.

Do you actually think it is possible to pick a single real number from a uniform distribution of continuous real numbers?

How would you go about implementing this experiment?

All real numbers have an infinite number of digits. Any that you can choose in your lifetime must have a finite number of non-zero digits, or be finitely expressible. Therefore, the decision process cannot be uniform since there is an inherent bias towards those that can be expressed in a human lifetime. However, there are only a finite number of reals with a finite number of non-zeros on the interval in question, and thus the probability of picking any one of them is no longer 0 but positive.

The Axiom of Choice is a misnomer. It is a mathematical existential claim completely unrelated to the notion of choice. It does not speak to whether it is possible to choose a particular random number uniformly from a continuous set of reals. If you think it does, construct a derivation.

Originally posted by DoctorScribbles
Do you actually think it is possible to pick a single real number from a uniform distribution of continuous real numbers?

How would you go about implementing this experiment?

All real numbers have an infinite number of digits. Any that you can choose in your lifetime must have a finite number of non-zero digits, or be finitely expressible. Th ...[text shortened]... number uniformly from a continuous set of reals. If you think it does, construct a derivation.

Picking a random real number uniformly between [0,1] is the same as picking one element each from a (countably) infinite set of sets, each of which is {0,1} - so it's a direct application of AC.

Your "proof" of the inevitability of sin depends on two assumptions - the equivalence of possibility and non-zero probability, and the immortality of Adam. Clearly, in a world where immortal souls exist, death is not a constraint on the experiment - and there is no limit to the number of digits that can be selected. So, this experiment can clearly be performed and it is possible to pick 0.5 even though the probability of that event happening is 0. Of course, this causes a contradiction - because of your axiom of the equivalence of possibility and non-zero probability.

Once again, it's a choice between two axiomatic worldviews - one with AC and one with the possibility-probability equivalence.

Originally posted by lucifershammer
If you want to reject AC because you want to maintain your axiom on the equivalence of possibility and probability, that's your decision. I don't draw the latter equivalence and I see no reason to reject AC.

I don't think possibility and probability are equivalent. Some things are improbable but not impossible.

I do think there is an analytical connection between possibility and probability.
If you don't, what meaning do you ascribe to events with probabilities of 0 or 1, if not impossible and certain?

Originally posted by lucifershammer
Picking a random real number uniformly between [0,1] is the same as picking one element each from a (countably) infinite set of sets, each of which is {0,1} - so it's a direct application of AC.

No it's not. The Axiom of Choice entails only the existence of some choice function. It does not guarantee that the choice function has any particular properties, such as corresponding to a uniform or equilikely distribution.

I don't need the Axiom of Choice to know that I could conceivably keep saying "1" repeatedly for the rest of my life. I have thus constructed a choice function without the aid of the Axiom of Choice, which can't construct anything more powerful even if I wanted to invoke its help. Thus, the Axiom of Chioce is irrelevant to this discussion. Even if I needed to use the Axiom of Chioce to know that such a function exists - that where 1 is repeatedly chosen - I do not have a function that corresponds to a process for picking a [0,1] real uniformly. Rather, my function picks .1 base 2 repeating with probability 1, which doesn't even approximate a uniform process.

You are grasping and latching on to something that you really don't understand, in a manner reminiscent of Coletti and his "Christian Logic."

Originally posted by DoctorScribbles
No it's not. The Axiom of Choice entails only the existence of some choice function. It does not guarantee that the choice function has any particular properties, such as corresponding to a uniform or equilikely distribution.

I don't need the Axiom of Choice to know that I could conceivably keep saying "1" repeatedly for the rest of my li ...[text shortened]... really don't understand, in a manner reminiscent of Coletti and his "Christian Logic."

Let's leave the insults to experts like no1, shall we?

You're right that AC does not guarantee a uniform distribution - but I don't actually need a uniform distribution in my example. All I need is the ability to pick any real number in [0,1] - and AC does give me that.

Originally posted by DoctorScribbles
I don't think possibility and probability are equivalent. Some things are improbable but not impossible.

I do think there is an analytical connection between possibility and probability.
If you don't, what meaning do you ascribe to events with probabilities of 0 or 1, if not impossible and certain?

Good question. I am increasingly beginning to subscribe to the view that probability is really a "plug" for lack of information about states of affairs.

Originally posted by lucifershammer
All I need is the ability to pick any real number in [0,1] - and AC does give me that.

OK. Tell me which choice function you are using and I'll tell you what the probability that you pick .5 is.

Originally posted by DoctorScribbles
OK. Tell me which choice function you are using and I'll tell you what the probability that you pick .5 is.

I believe I did describe one a few posts back (timestamp 19:52).

Originally posted by lucifershammer
I believe I did describe one a few posts back (timestamp 19:52).

No you didn't. You described some properties of one, but it is not completely specified.

How do you pick each digit?
How many digits do you pick?


Picking a random real number uniformly between [0,1] is the same as picking one element each from a (countably) infinite set of sets, each of which is {0,1}

That's right. They're both impossible.

Originally posted by DoctorScribbles
No you didn't. You described some properties of one, but it is not completely specified.

How do you pick each digit?
How many digits do you pick?


[b]Picking a random real number uniformly between [0,1] is the same as picking one element each from a (countably) infinite set of sets, each of which is {0,1}

That's right. They're both impossible.[/b]

Saying that's impossible is axiomatic (i.e. it is the acceptance of ~AC) - just as saying it's possible is axiomatic (AC).

As I said earlier, it's a clash of two axiomatic worldviews.

Originally posted by lucifershammer
Saying that's impossible is axiomatic (i.e. it is the acceptance of ~AC) - just as saying it's possible is axiomatic (AC).

As I said earlier, it's a clash of two axiomatic worldviews.

Then specify a process to pick the number.

How do you pick each digit?
How many digits do you pick?

Originally posted by DoctorScribbles
Then specify a process to pick the number. You can't do it.

I don't need to. You're deliberately asking for a constructivist solution in a non-constructivist framework. Can you prove that such a process does not exist?

Originally posted by lucifershammer
Saying that's impossible is axiomatic (i.e. it is the acceptance of ~AC) - just as saying it's possible is axiomatic (AC).

As I said earlier, it's a clash of two axiomatic worldviews.

You keep confusing two things.

To say that it is impossible to uniformly pick a real is not to deny AC.
To say that it is impossible to pick a real is to deny AC.

Are you claiming that you can uniformly pick a real, or that you can merely pick a real? I'm claiming you can't do it uniformly, even if you accept the AC.

If you're not claiming your process is uniform, then I don't doubt its existence. Just tell me its specifications and I will tell you the probability of picking .5 under it.