You're right, the limit of that set is 1, which is one of the proofs. But you can say 0.9recurring = 1, they are two ways of writing the same number in decimal form. Any decimal can have an infinite number of digits. But really it is the limit of a geometric series, yes.

(April 23, 2015 at 9:51 am)robvalue Wrote: But you can say 0.9recurring = 1, they are two ways of writing the same number in decimal form...

Why, shore! Let's define 0.999999.... = 1.  I've heard computers have headaches with +0 and -0 as well. But that's another story.  

(April 23, 2015 at 9:43 am)Hatshepsut Wrote:

(April 18, 2015 at 10:01 pm)noctalla Wrote: A thought experiment consisting of two scenarios and follow up questions:

Scenario A: In this scenario, we have an apparently omniscient Being. As far as the Being is aware, it knows and understands everything that there is to know and understand. It is also the case, in this scenario, that the being happens to be correct that it knows and understands everything there is to know and understand.

Scenario B: This scenario is identical to the first in, insofar as we have an apparently omniscient Being that appears to know and understand everything there is to know and understand. The only difference is, that in this scenario, the being is wrong.

Question: How could an omniscient Being determine which of the two scenarios it was actually in?

The OP problem and probably nontrivial. The SEP (Stanford Plato) definition for omniscience was that the Being knows every true proposition. If, for each falsehood, there is a corresponding true proposition asserting it is false, then the Being knows the truth value of every proposition as well. However, that leaves out the possibility of propositions that do not have a truth value. Now let Q be the proposition that the Being can  determine whether it is in Scenario A or B, regardless of how it does so. Does Q have a truth value?

I tend to think there are big problems with absolute omniscience or omnipotence. It's a big leap from Apollo knowing a hell of lot and having power to throw lightning bolts to the "omni" stuff. These ideas weren't part of the "original package" of religious belief even for the Christian deity, but introduced much later by thinkers like Augustine and Thomas Aquinas. That's despite the "fall of every sparrow" and "number of hairs on your head" bible ditties. Knowing those two and similar things isn't  knowing everything. The good/evil issues like babies that shrivel up are a separate problem, the "problem of evil," related to omnibenevolence, I think.

• (BTW Also unsure if 0.999999... = 1 is ever claimed. I thought it was that the sequence S = {0.9, 0.99, 0.999, ... } has a least upper bound, namely 1. But none of the individual sequence members are equal to 1, they're all less than 1, and the object 0.999999.... may not exist. I dunno, I'm hardly pluripotent much less omni.   )
  

I suspect that Q does have a truth value and that value is untrue. If that's the case, the SEP definition for omniscience is paradoxical (which is my original point: omniscience, if you impose a very strict definition such as the SEP, is an impossible standard). It results in paradox because whether or not the Being is in Scenario A or B also has a truth value, and so an omniscient being would have to know something that it knows it cannot know (it reminds me of the Liar Paradox e.g. "this sentence is false"). However, if I'm incorrect and it turns out Q is true, then omniscience is preserved - but I know of no proposed solution to the problem that results in Q being true.

Omniscience and omnipotence are such childish claims.

Can you picture this conversation between children?

"My hero is really powerful!"

"How powerful?"

"Like, so powerful! He made the whole world!"

"Wow! The whole world? What else can he do?.

"Erm... he can travel in time!"

"Cool! Well my super hero can do all that, and he made the whole universe!"

"He did? Well... my super hero made your super hero! In fact... my super hero can do anything! Beat that! And he knows everything!"


Even if "God" made the universe, it does not follow that he can do "anything". Nor does it follow that he knows everything. They are ridiculous claims plastered on to a being whose merest existence cannot be demonstrated in the first place, and who is defined in such a way that we can know nothing about him.

The fact that I'm saying "him", because the bible gives God a human male gender, is a bit if a giveaway. Oh, and lots of human emotions. And he walked on the earth like a human. He used to do magic tricks, but not so much recently.

(April 24, 2015 at 11:03 pm)noctalla Wrote:

(April 23, 2015 at 9:43 am)Hatshepsut Wrote: Now let Q be the proposition that the Being can  determine whether it is in Scenario A or B, regardless of how it does so. Does Q have a truth value? (See OP)

I suspect that Q does have a truth value and that value is untrue. If that's the case, the SEP definition for omniscience is paradoxical (which is my original point: omniscience, if you impose a very strict definition such as the SEP, is an impossible standard). It results in paradox because whether or not the Being is in Scenario A or B also has a truth value, and so an omniscient being would have to know something that it knows it cannot know (it reminds me of the Liar Paradox e.g. "this sentence is false"). However, if I'm incorrect and it turns out Q is true, then omniscience is preserved - but I know of no proposed solution to the problem that results in Q being true.

I think your bolded part is a premise in the OP which we have to grant, or the whole OP setup would be pointless. Good. That clearly implies the Being cannot be in both A and B. Now we can dispense with Q = true. Suppose the Being is in scenario B. Then the Being thinks that it is in A, because it's wrong. In this case, the Being fails to determine its scenario, so Q cannot be true. We haven't yet shown that Q = false, because while it's false provided the Being is in B, we still need to show it is false if the Being is in A. I haven't been able to do that.

Your theorem that Q = false ---> The SEP definition is paradoxical ...?? I'm stumped. One half of it is easy, like you said. Let Being satisfy the SEP. Then it knows that Q = false. Therefore it doesn't know if it's in A or B. Therefore it doesn't satisfy the SEP.

But to finish paradox, we must let Being not satisfy SEP and show this implies that it does satisfy SEP. If we have only the first half, it's possible the class of Beings which satisfy SEP is empty.

(April 25, 2015 at 3:32 am)robvalue Wrote: Omniscience and omnipotence are such childish claims.

Kind of like angels dancing on a pinhead.

@Hatshepsut - My brain keeps going around in circles trying to resolve the problem. I'm going to have to stop thinking about it (if I can) before I get dizzy.

@robvalue - Omniscience and omnipotence are rather silly. As you describe, it sounds like the end point in a game of religious one-upmanship. Unless there's a way to one-up omniscience/omnipotence...

Flanders: Does not!
Homer: Does too!
Flanders: Does not!
Homer: Does too infinity!
Flanders: Does not infinity plus one!
Homer: Doh!

Sure, there is something more powerful and more knowledgeable than an omnipotent and omniscient being. All God has to do is make it. He can do that, right?

(April 26, 2015 at 2:29 am)robvalue Wrote: Flanders: Does not!
Homer: Does too!
Flanders: Does not!
Homer: Does too infinity!
Flanders: Does not infinity plus one!
Homer: Doh!

Sure, there is something more powerful and more knowledgeable than an omnipotent and omniscient being. All God has to do is make it. He can do that, right?

From one Pandora's Box to another.

(April 26, 2015 at 2:25 am)noctalla Wrote: @Hatshepsut - My brain keeps going around in circles trying to resolve the problem. I'm going to have to stop thinking about it (if I can) before I get dizzy.

@robvalue - Omniscience and omnipotence are rather silly....

I'm not going to get any farther with it either, I'm not good enough at it. But oddly, I've never seen any real proofs one way or the other related to omniscience and omnipotence, and suspect that has something to do with our inability to define these terms adequately. They're about as silly as thousands of angels dancing on a pinhead.