It’s obvious that you never bothered to read my exposition on Gödel’s Incompleteness Theorem, let alone attempt to understand said exposition, though you 've set several sordid precedents in this vein in the past, where you blithely ignored what I posted, and continued posting your ignorance-driven apologetic fabrications.
This, time, READ my fucking post, and exert an effort to UNDERSTAND what I’m telling you.
When Gödel constructed his Incompleteness Theorem, he constructed a special function, which I represented above symbolically as S(x). This is a Boolean function, which is true if and only if x is the Gödel number of, wait for it, a proposition that is both true AND provable in the formal system under consideration. Gödel’s master stroke consisted of proving that there existed a special number p, such that ~S(p) was true. This statement said that p is NOT the Gödel number of a proposition that is both true and provable in the formal system of interest. I emphasise this point again for a good reason. However, lo and behold, p happens to be the Gödel number of ~S(p) itself.
The conclusion arising from this is that ~S(p) is either false, in which case the formal system is inconsistent, or else, ~S(p) is true, in which case ~S(p) is an example of a true proposition that is not provable within the formal system of interest.
This of course doesn’t mean that ~S(p) cannot be proven in a different formal system. But the point here is that undecidable propositions are still true propositions, they are merely not provable to be true in the current formal system. As a corollary, as I stated above, sophisticated mathematicians use this as a tool for cracking certain hard problems - namely, if you can demonstrate that your proposition is Gödelian-undecidable in your current choice of formal system, then it is still a true proposition.
Now, dealing with your tedious and ignorant apologetic brainfart:
The whole point being, idiot, that the proposition of interest will NOT be undecidable in that other choice of formal system. Are you really too stupid to realise this fact, given that I explicitly presented it in my previous exposition?
Once again, if doing so solves the problem for the current proposition of interest, so fucking what?
Are you being deliberately fucking obtuse here?
Oh, and Russell’s recognition of the ramifications of Gödel’s Incompleteness Theorem, was almost certainly accompanied by a level of thought far surpassing anything you’ve exhibited here.
Once again, exert the effort to LEARN WHAT PEOPLE HERE ARE TEACHING YOU, instead of ignoring their efforts and continuing to plough on regardless with your bullshit apologetics.
No. Atheism is built on non-belief in God’s claims. Many atheists happen to be rationalists, scientists, skeptics and others. Not all. Some are Buddhists, New Agers, Satanists, and believe in all sorts of silliness. Perhaps your problem is that you do not have a solid definition of ‘Atheist.’
EDIT: I should have read Sheldon’s post prior to commenting. I thought I was at the end of the thread. Sorry for being repetitious.
Evidence of the non-existence of God is that I see nothing resembling God in the Universe, for example. It has been proven that the Universe is expanding indefinitely into cold death, what then can you say?
Thank you for your repetitive unsubstantiated nonsense. Repeating bullshit claims while ignoring those who have posted to try and help you straighten out your thinking, will eventually get you booted from the site. There is a distinction between an anti-theist and an atheist.
Anti-theist makes the claim - ‘No god exists.’ Or quite possibly ‘Your god does not exist.’ My guess is this: If you have run into these sorts of atheists often, your arguments for the existence of god are shit. Many atheists will take an anti-theist position against a god claim made of pure ignorance. Have you looked at your god claims lately?
Atheists are simply people who do not believe in god or gods. For these people, it is usually the lack of evidence for god claims that motivates them. The burden of proof is on the person making the claim. Have you any good evidence for your god claims?
All anti-theists are atheists but not all atheists are anti-theists. Would you like a Venn Diagram? You know, there is a clear illustration of this on the Home Page. I am sorry you didn’t understand it before logging into the site. I am sure someone would be happy to take the time to explain it to you. Just, Not Me. I am allergic to whiny puppies of the female persuasion and don’t want to risk a case of hives.
LOL, Internet. I’ll take Brian’s story over yours any day of the week. Besides, why not admit it! You don’t have a clue what ‘proper science’ means. You couldn’t even get the gasoline question right.
Okay, okey. … Two penguins are in the bathroom and the first penguin turns to the second penguin and says, ‘Hey. pass me the soap.’ Now, what kind of soap was it?
Some do, but many do not, and I am in the latter category, as you have been told.
Indeed, but this is insufficient to claim no deity exists, you didn’t even define any deity for a start, this is very poor reasoning.
No it hasn’t, science doesn’t really prove anything, it gathers evidence and creates hypothesis, then test these models against reality, if the evidence supports the hypothesis and the predictions it makes matches reality, then when that evidence becomes conclusive it may reach the pinnacle of scientific endeavour an accepted scientific theory, which explains why aspects of the natural world and universe behave as they do, within such theories may be scientific laws, that explain how very specific smaller parts of the theory behave. All scientific ideas, even accepted irrefutable facts like species evolution for example, must remain tentative in the light of new evidence no matter how unlikely this prospect is.
Are you sure?
That YouTube link is from Dr Brain Cox, CBE, he’s a professor of particle physics in the School of Physics and Astronomy at the University of Manchester and The Royal Society Professor for Public Engagement in Science. Now if you have credentials in Physics then do please cite any published articles of yours in the field? Until then, given I know who Dr Cox is, and you don’t seem to understand basic scientific terminology, I shall be siding with @Cognostic and Dr Cox.
I am not even sure that we are disagreeing at this point about Godel’s IT. I cannot formally prove it but I understand the point of the IT in this simple statement: “This statement is false.” This shows how mathematical logic is incomplete or inconsistent.
And are you agreeing with Russell’s recognition and ramification of IT?
If you cannot prove something to be true what do you employ to make it a truth statement?
If you can’t prove a statement to be true, then you just don’t use the statement to make mathematical deductions. Your notion that this is inconsistent is extremely misguided (at best, my guess is it is willful ignorance). It is practically the opposite of inconsistency.
It is almost as if you want mathematics to be inconsistent; so when you encountered something that you thought said that, you’ve incorporated that falsehood into your worldview. If you care about accuracy, you need to get rid of the notion that mathematics is inconsistent; that is very wrong, and marks you as someone who doesn’t have a clue what they are talking about.
Once and for all, WHEN ARE YOU GOING TO STOP PLAYING DISHONEST APOLOGETICS WITH THE FACTS?
Let’s deal with this once and for all, shall we?
When David Hilbert, before his death in 1922, included the question of the completeness of elementary number theory in his grand list of outstanding questions for mathematics, he did so in a precisely defined manner. The 23 problems that have since been labelled “The Hilbert Programme” . Problem #2 of that programme was stated as “Prove that the axioms of elementary number theory are consistent”. It was, of course, realised at the time, not merely by Hilbert, that success in this endeavour would mean, given the state of art of mathematical knowledge in 1900, when the Hilbert Programme was launched, establshing that elementary number theory was complete and decidable, concepts that have two precise definitions, viz:
[1] A formal system is complete, if every TRUE proposition of the system can be proven therein (note my emphasis on the word TRUE here);
[2] A formal system is decidable if there exists a mechanical procedure for deducing the proofs covered in [1] above.
Note, and I emphasise this STRONGLY, that we are dealing here [balways[/b] with TRUE PROPOSITIONS.
Indeed, no less a person than Willard van Ormand Quine, the author of one of the 20th century’s seminal textbooks on logic, covered this very topic in some detail, from the end of page 243 to 245 onwards, viz:
I’ll leave out several subsequent paragraphs, because ultimately this leads to the very same exposition of Gödel’s master stroke on page 246, that I provided in my previous post, though using different notation than Quine’s.
The point I am ramming home here, and which I repeatedly stated above, is that UNDECIDABLE PROPOSITIONS ARE, BY CONSTRUCTION, TRUE PROPOSITIONS. Just because they cannot be proven true in the current formal system of interest, does NOT mean for one moment that they cannot be proven true in a different formal system, a point you duplicitously ignore from my previous exposition. Indeed, since Gödel’s attack will of necessity involve a different Gödel numbering scheme for each different formal system, the undecidable propositions for two formal systems of interest will be different, (with at worst only occasional overlap) and as a corollary, it will always be possible for mathematicians to find a new formal system, withiin which a proposition undecidable in the current formal system is provable. Another point you continue mendaciously to ignore in your dishonest apologetics.
Indeed, I know exactly why you’re pulling this blatant piece of dishonesty, and your modus operandi here is as transparent as it is duplicitous. You’re hoping to press Gödel’s Incompleteness Proof into dishonest apologetic service, to propagate the bullshit idea “Mathematics is inconsistent, therefore mathematics and science can’t be trusted, therefore Magic Man”. It’s so blatant a piece of apologetic fabrication, that I don’t even need to have spent 14 years dealing with yourilk to recognise it as such, though of course said past experience helps.
The above destroys this deliberate and mendacious apologetic twisting of my previous expositions wholesale.
Now drop the duplicity once and for all, mythology fanboy.
Oh, by the way, a different approach was taken ultimately to establish the consistency of the axioms of elementary number theory, courtesy of Gerhard Karl Erich Genzten, who in 1936, published a proof to the effect that said consistency follows as a corollary of the well-foundedness of the ordinal ε0. the 1936 paper covering this is the following:
Die Widerspruchsfreiheit Der Reinen Zahlentheorie by Gerhard K. E. Genzten, Mathematische Annalen, 112: 493-565 (December 1936)
Indeed, it transpires that I was able to download the entire December 1936 issue of Mathematische Annalen, courtesy of this link from the university of Göttingen, where Gentzen spent a considerable part of his career.
Sadly, my O-level Greman isn’t up to translating that paper, and I don’t have an English tranlation to hand, but his basic argument is that the Peano axioms of elelentary number theory cannot generate a proposition informing us of the well-formedness of the ordinal ε0, ansd as a corollary, that the Peano axioms form an incomplete system. However, it transpires that there exists an independent proof of the well-formedness of ε0. Because a proposition known to be true by independent proof is undecidable via the Peano axioms, those axioms form a consistent system.
I will quote the following because I honestly think it sums up Godel’s results well.
Penrose said: “The inescapable conclusion seems to be: Mathematicians are not using a knowably sound calculation procedure in order to ascertain mathematical truth. We deduce that mathematical understanding – the means whereby mathematicians arrive at their conclusions with respect to mathematical truth – cannot be reduced to blind calculation!”
Russell said: “I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure then the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.”
Can you demonstrate any objective evidence for any deity or deities? The certainty of religious faith, unless you can demonstrate something beyond a bare claim, merely donates a closed mind.
NB My atheism is simply the lack or absence of belief, it doesn’t carry therefore any epistemological burden of proof, and certainly does not require absolute certainty, nor do I believe such a thing is even possible.
I genuinely have difficulty coming to the conclusion that anyone can say they don’t believe in absolute certainty. One cannot make that claim and not self refute. Because the moment one says “I don’t believe in absolute certainty.” This is self refuting. If one says such a statement then it turns around and places uncertainty in the statement said, leaving room in believing in certainty but that refutes the statement. So that when it’s true it’s false and when it’s false it’s true.
