txaq007's Inescapable Error

[chezlaw]

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"I am not well versed in mathematical proofs, so I am wondering from others on the site if there are indeed mathematical certainties that cannot be proved. Anyone?"

No. Period.

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Hmm, very interesting David...

How do you explain something like Godel's incompleteness theorems? And I guess you then know that i.e. the Goldbach conjecture and the Riemann hypothesis are provable or are they not mathematical certainties?

And I guess the continuum hypothesis is then a mathematical certainty? Or how does that work?

For someone who keeps insisting on the power of mathematics and logic it seems you know very little about these disciplines.

- Kripke

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Riemann and Goldbach remain uncertain until proved or disproved. If they are true but unprovable then they remain forever uncertain.

The continuum hypothesis is independent of the standard axioms and can be taken as an additional axiom or not.

What's the problem?

The only smallish problem is that maths cannot be proved to be consistent and so every theorem could be provable along with its negation, but we don't worry about that [img]/images/graemlins/smile.gif[/img]

chez

[Kripke]

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"I am not well versed in mathematical proofs, so I am wondering from others on the site if there are indeed mathematical certainties that cannot be proved. Anyone?"

No. Period.

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Hmm, very interesting David...

How do you explain something like Godel's incompleteness theorems? And I guess you then know that i.e. the Goldbach conjecture and the Riemann hypothesis are provable or are they not mathematical certainties?

And I guess the continuum hypothesis is then a mathematical certainty? Or how does that work?

For someone who keeps insisting on the power of mathematics and logic it seems you know very little about these disciplines.

- Kripke

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Riemann and Goldbach remain uncertain until proved or disproved. If they are true but unprovable then they remain forever uncertain.

The continuum hypothesis is independent of the standard axioms and can be taken as an additional axiom or not.

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Yet, the continuum hypothesis CAN be proven on the basis of standard ZFC axioms. That it can also be refuted on the basis of ZFC is the problem here.

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The only smallish problem is that maths cannot be proved to be consistent and so every theorem could be provable along with its negation, but we don't worry about that [img]/images/graemlins/smile.gif[/img]


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Exactly, let's not worry about such minor details. [img]/images/graemlins/wink.gif[/img]

- Kripke

[chezlaw]

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Yet, the continuum hypothesis CAN be proven on the basis of standard ZFC axioms. That it can also be refuted on the basis of ZFC is the problem here.

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This is beyond my level of expertise but I though Godel proved that ZFC+CH was consistent and someone else proved that ZFC+~CH was consistent.

I've never heard of any proof that ZFC->CH (or ZFC -> ~CH). Sounds unlikely as then ZFC would be proven inconsistent.

I'm in over my head so I'll read what others have to say.

chez

[BluffTHIS!]

All you religious believers who have debated with David in this and other threads are a bunch of rubes. You let him frame the question and then only try to rebut that without turning the tables. If you analyzed the basis of all David's reasoning on religion you would find that it based on a core premise, namely that the question of the truth or falsity of religious belief should be viewed as a probability question. And even if you are willing to admit as I am for the sake of arguement that it can be viewed as such, there is the much larger question of what is the utility of such probabilities to your beliefs and life, especially if the question is brought up of the scale of consequence if his probabilities turn out to be false, i.e. the longshot comes in, or of how great is the cost to take the dog view. A related question of how useful such probabilities might be is whether they are viewed or not in the context of a game such as poker, that has a positive or negative expectation overall as well as for various "plays". He and I have exchanged a few posts on this in the past and disagree, but nonetheless, these are the only avenues by which you can attack David's lines of reasoning on religious questions but you all never look deep enough to see that.

You all are just philosophical, religious and logical donkeys.

[Kripke]

Godel proved that the CH was consistent with the axoims of ZFC, but conjectured that it could probably be proved inconsistent as well depending on what axiom of infinity you assume. This was the proven some years after Godel's conjecture.

In other words, the CH is demonstrably unprovable on the basis of the Zermelo-Fraenkel axioms.

Maybe I phrased myself wrong. The zermelo-fraenkel axioms have of course NOT been proven inconsistent.

- Kripke

[chezlaw]

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Godel proved that the CH was consistent with the axoims of ZFC, but conjectured that it could probably be proved inconsistent as well depending on what axiom of infinity you assume. This was the proven some years after Godel's conjecture.

In other words, the CH is demonstrably unprovable on the basis of the Zermelo-Fraenkel axioms.

Maybe I phrased myself wrong. The zermelo-fraenkel axioms have of course NOT been proven inconsistent.

- Kripke

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Okay we're saying the same thing now. So none of Riemann, Goldbach or CH provide any problem to DS's claim that there are no unprovable certainties in maths.

That just leaves the consistency problem but is this really a problem. If we have some theorem T such that ZFC->T then we are not claiming certainty that T is true, but are claiming certainty that ZFC->T which is true whether or not ZFC is consistent.


chez

[txag007]

"You are right as usual. But his original statements implied that his 100% certainty relied only on arguments he was espousing."

That implication was not intended. I've said in previous posts that if you are waiting for proof beyond a shadow of a doubt before you believe, you'll be waiting a while. There is enough evidence, I've said, to be reasonably sure Christianity is correct. I've tried to present that evidence on these boards. Again, if you implied something different from yesterday's posts, I apologize.

[txag007]

"Can there be an experience that conveyes the divinity of god? Assuming you have some experience how do you know it wasn't caused by a non-divine god using some advanced science?"

The excuses one can make to avoid believing are infinite in number. That question cannot be answered from a non-Christian perspective, other than looking at the fact that there is absolutely no evidence for that.

[txag007]

My question remains: Have you, yourself, taken an objective look at the evidence for Christianity?

[Kripke]

But there is a problem chezlaw.

Assume for the sake of argument that the Goldbach conjecture is true. If it is true, then it is also certain.

Mathematical truths are not contingent truths such that they can be true, but then turn out to be false later on (if they turn out to be false, then they were never true). They are necessarily true (i.e. true in every possible world). Thus when David proclaims that every mathematical certainty can be proved he is simply making a stipulation for which he has no proof nor argument.

It might very well be the case that the Goldbach conjecture is unprovable, but that does not mean that it is not a mathematical certainty. Now, from an epistemological point of view, we may never acquire certainty of the truth of this hypothesis, but that does not entail that it is not certain.

Godel's incompleteness proof is just a much better illustration of this fact. In that proof it is exactly proved that any mathematical theory containing arithmetic will contain propositions which are entailed by the axioms of the theory, yet not provable on the basis of the theory.

Since these propositions can be proved by adding a meta-theory, these propositions are certainly certain, yet they are not provable within the theory.

My basic point was just that David once again is making stipulations which he is entirely unable to prove, justify or even give a sound argument for.

- Kripke