Explain Godel's Ontological Proof in layman's terms?
439110
When I call something a fact it is not really a fact but rather an assumed fact.\n\nFact: God is defined as being greater than anything that can be imagined. ( The defined as is an important part of the proof)\n\nFact: God exists in understanding. (a person can imagine that a god exists)\n\nFact(dubious): If God exists in understanding, We can imagine him as greater if he exists in reality. ( The idea is if you imagine the great god as only existing in our imagination, and then imagine him existing in reality He would be greater in the case where he exists in reality.)\n\nNow here is the squeeze play. Go back to the definition of god as being greater than anything that can be imagined. Now of the two possibilities, that god exists only in our imagination, and God exists in reality, only the later that god exists in reality fits the definition of God, Therefore according to this logical construct God Must exists. I like to call this a logic box. You build parameter that insures an outcome that God exists. \n\nThe problem shows up in that are all the facts real, and also just because something can be logically shown to fit a definition is the definition true.
495947
i am intrigued by your idea of LOGIC!!! interesting
818651
(a) Suppose f is NOT injective, that is, f( x') = f(x'') for some x' and x'' in X. Then also g(f( x')) = g(f( x'')), which means that g o f is NOT injective. (This is proof by transposition: instead of showing directly that A implies B, we show that Not B implies Not A.)\n\n(b) Take any t in T. Then t = g(f(x)) for some x (since g o f is surjective). This means that t = g(y) for some y, namely, for y = f(x). So g is surjective.\n\n(c) By Part (a), f is injective. By Part (b), applied to f o g in place of g o f, f is surjective. Beig both injective and surjective, f is bijective.
7873
well in the demostrantion of the Real numbers...\n\nall the numbers and relations depend of the existence of 1... suck that there are some rules..\n\nso we even demostrate that 1 exists... and that 1 > 0... after that we can prove the existence of any other Real number..
246532
Read 'The Unity of Philosophical Experience' by Etienne Gilson.  He ably dismantles this kind of Cartesian nonsense in the relevant section.
637043
i believe what you mean are "existential" and "universal" claims. the word existential refers to words such as some, many, few, or any other word which implies the existence of another set which does not contain members for which the "some..." statement is referring to. a universal statement includes the other group of words including all, none, every, etc which do not imply the existence of a group without the attribute referred to by the statement containing the all, every, etc.\n\nthe problem with universal statements involves a "universal generalization" about an entire set based on the existence of a few members which meet the criterion. example. "my dog is red, joes dog is red too, therefore all dogs are red."\n\nthe problem with the existential statements involve "existential instantiation" let me give you an example. "some dogs are red, i have a dog, therefore he is red". "if it rains the sidewalk is wet. the sidewalk is wet so therefore it rained." you see the problem with this line of reasoning\n\nin short \ndont conclude "all a is b" based on "a0 is b"\ndont conclude "a0 is b" based on "some a is b"
113690
No.  Deductive proofs start from a "given", that is, a fact accepted as true or a theorem previously proven to be true.  But other kinds of proofs start with an assumption that the theorem to be proved is not true, and from that assumption derive some contradiction, thus proving that the theorem itself is true.\n\nAn example would be the proof that there is no highest prime number, which goes like this. Assume that there is a highest prime number.  Call it x.  Now consider the number x! + 1.  (X! is X times X-1 times X-2 all the way down to 1.)  Since x! is divisible by 2, x! + 1, being just 1 more than x!, is not divisible by 2.  Similarly, x! + 1 is not divisible by 3, 4, or any other integer up to and including x.  Since x! + 1 is not divisible by any number up to x, either it is prime, or it's prime factors are larger than x.  But that contradicts our assumption that x is the largest prime.  Thus, our assumption is false.  QED.
546909
Definitions of the terms. The axioms for the natrual numbers are as follows:\n\n1) 0 is a natural number,\n2) The successor of a natural number is a natural number,\n3) No two numbers have the same successor\n4) 0 is not the successor of any natural number\n5) If P(x) is a property of natural numbers such that P(0) is true and such that P(n) true implies P(succ n) is true, then P(n) is true for all natural numbers (Mathematical Induction)\n\nUse S(n) to denote the successor of n and *define*\n1=S(0) and 2=S(1).\n\nNow *define* addition inductively by\nn+0=n and\nn+Sm=S(n+m).\n\nThen,\n1+1=1+S(0)\n=S(1+0)=S(1)=2.
348468
This hypothesis is false:\nSuppose 0/0 = 1. Since 1 is the multiplicative identity, this implies (0/0)*a = a for all real numbers a. But (0/0)*a = (0*a)/0 = 0/0 = 1. This contradicts our hypothesis, so the hypothesis must be false (proof by contradiction). QED\n0/0 is indeterminate as a limiting form. See http://mathworld.wolfram.com/Indeterminate.html . \nAs an algebraic expression over the real field, it is undefined: http://mathworld.wolfram.com/Undefined.html . \nThis is a Frequently Asked Question; the following is my previous answer:\n\nThis question may be asked so frequently (along with the infamous 0.999...) because no concrete axioms or definitions are ever given for the real number system until one studies higher algebra or real analysis; it is assumed that everyone knows what a real number is until then. The proliferation of these questions show that this assumption is no good.\nFor the sake of conciseness, I will not define what a real number is here, as only the algebraic structure of the real numbers is necessary to answer this question. The algebraic structure of the real numbers is what we call a field. A field is an algebraic structure that follows a finite set of axioms. These axioms define what statements are true and false in this system based on the laws of logic. You can find the field axioms here: http://mathworld.wolfram.com/Field.html .\nEach line is preceded by "For all real numbers a, b and c, the following is true...". Note that the proper form of the additive identity and multiplicative identity is that it prescribes the existence of some number 1 and some number 0 such that the equations hold. These same numbers are then used to prescribe the existence of additive and multiplicative inverses.\nDivision a/b is just shorthand for a*(b^(-1)). So in your question, you are looking for 0*0^(-1). It is easy to show that there is no such number as 0^(-1) in any field. Suppose 0^(-1) exists.\nThen 0*0^(-1) = 1. But multiplying both sides by 0^(-1) gives 0 = 0^(-1), which means 0*0 = 0 = 1. This is obviously false.\nSuppose 0 = 1. 0*a = a for all numbers a. But 0*a = (1 + (-1))*a = a + (-a) = 0. Thus 0 is not equivalent to 1 and our sole assumption that 0^(-1) exists must be false. Tus the expression 0*0^(-1) is undefined in a field.\nProof by contradiction is very powerful for many types of statements.\nAs to the limit argument, note that you are evaluating a limit over a plane of the function f(x,y) = x/y. As we approach (0,0) along the x-axis, this limit is 0, but as we approach (0,0) along the y-axis, the limit diverges. Thus, the limit of f(x,y) at (0,0) does not exist, so f(0,0) remains undefined with no continuous way to complete it. The limit is not indeterminate for this function; it does not exist.\nThe expression 0/0 is said to be indeterminate as a limiting form because one can have many functions that approach the form 0/0 that have convergent limits of different values. For example, the limit as x approaches 0 of [sin(x)]/x yields the indeterminate form 0/0, but L'Hopital's rule reveals the limit to be 1. The expressions listed on this page: http://mathworld.wolfram.com/Indeterminate.html are called indeterminate because they do not have a fixed behavior when studied as limits of expressions.
597619
well that last example doesnt work...they divide by zero and try to let it slip.\n\nThings like Russells paradox messed with the mathematical world for a while.  This paradox deals with sets.\n\nIt says that we have a set R, R is the set of all sets that are not members of themselves.  This means that R  is neither a member of itself nor not a member of itself.\n\nThis is pretty neat, and it was a huge problem with set theory.  It was solved by saying that all sets are members of a universe.  I still claim that its a sticky area.\n\nhope this is what you were looking for.\n\nMatttlocke
212175
:}  No one can know anything for a fact beyond any doubt..\n\nEven our very senses fool us. There are dreams and mirages. We’ve all thought we heard saw or smelled something that was inaccurate.\n\nThoreau in Walden Pond, Presented the idea that we could know that we exist because if we did not we could not question whether we did or not.\n\nHaving established that nothing, beyond perhaps our own existence, can be established with absolute certainty, we must proceed to behave as if those things for which there is significant evidence, in fact exist and work as the evidence would suggest they do.\n\nWhile this procedure at times will lead to error, for the most part it is necessary to pursue this line of reasoning, called inductive logic, if we are to conduct normal life functions.\n\nIs there a god?  I use the same process. So for my purposes Yes, as far as my data goes in allowing  me to analyze the information, the preponderance of evidence suggest that God is a certain fact.  \n\nThen using the scientific method it simply remains to see if the predictions I make based on this hypothesis come to be\n\nAnd they have.\n\nI'm 100% sure \n\n\n\nOne additional fact \nMy salvation depends less on my faith than on his faithfulness!
906336
Absence of proof is not proof of absence.  \n\nHowever, it is the logical (or I should say illogical) fallacy "argument from ignorance" that because there is no evidence that God exists, that is evidence that God does not exist.   \n\nIt is also logical fallacy of "negative proof," that God must exist because there is no proof that God does not exist.  Why do you think God exists.  Why do you think God doesn't exist?    \n\n"The burden of proof should be on the proposed idea, not the challenger of the idea."  The challenger must only raise an issue of doubt, whereas the proposer must carry the burden of persuasion.  \n\nWhat can be more "honest," "practical," "intelligent," than LOGIC?\n\nAnother point, it is beliefs we are talking about, which are devloped best from the weight of the evidence in a world that seldom presents absolute proof.  Provocative question!  Thanks for keeping me up all night with this non-sense!
800729
Whenever we argue about whether a thing can be proved, we should distinguish five different questions about that thing:\n\n\nDoes it really exist or not? "To be or not to be, that is the question." \nIf it does exist, do we know that it exists? A thing can obviously exist without our knowing it. \nIf we know that it exists, can we be certain of this knowledge? Our knowledge might be true but uncertain; it might be "right opinion." \nIf it is certain, is there a logical proof, a demonstration of why we have a right to be certain? There may be some certainties that are not logically demonstrable (e.g. my own existence, or the law of non-contradiction). \nIf there is a proof, is it a scientific one in the modern sense of 'scientific'? Is it publicly verifiable by formal logic and/or empirical observation? There may be other valid kinds of proof besides proofs by the scientific method. \nThe fifth point is especially important when asking whether you can prove life after death. I think it depends on what kinds of proof you will accept. It cannot be proved like a theorem in Euclidean geometry; nor can it be observed, like a virus. For the existence of life after death is not on the one hand a logical tautology: its contradiction does not entail a contradiction, as a Euclidean theorem does. On the other hand, it cannot be empirically proved or disproved (at least before death) simply because by definition all experience before death is experience of life before death, not life after death.\nIf life after death cannot be proved scientifically, is it then intellectually irresponsible to accept it? Only if you assume that it is intellectually irresponsible to accept anything that cannot be proved scientifically. But that premise is self-contradictory (and therefore intellectually irresponsible)! You cannot scientifically prove that the only acceptable proofs are scientific proofs. You cannot prove logically or empirically that only logical or empirical proofs are acceptable as proofs. You cannot prove it logically because its contradiction does not entail a contradiction, and you cannot prove it empirically because neither a proof nor the criterion of acceptability are empirical entities. Thus scientism (the premise that only scientific proofs count as proofs) is not scientific; it is a dogma of faith, a religion.\n\n\nI.\nThe first reason for believing in life after death is simply that there is no compelling reason not to, no objection to it that cannot be answered. The two most frequent objections are as follows:\n(a) Since there is no conclusive evidence for life after death, it is as irresponsible to believe it as to believe in UFOs, or alchemy. Perhaps we cannot disprove it; a universal negative always is difficult if not impossible to disprove. But if we cannot prove it either, it is wishful thinking, not evidence, that makes us believe it.\n\nNow this objector either means by 'evidence' merely empirical evidence, or else any kind of evidence. If he means the latter, he ignores all the following proofs for life after death. There is a lot of evidence. If he means the former, he falls victim to the self-contradiction argument just mentioned. There is no empirical evidence that the only kind of evidence we should accept is empirical evidence.\n\nIn most supposedly scientific objections of this type, an impossible demand is made, overtly or covertly-a demand for scientific proof-and then the belief is faulted for not satisfying that demand. This is like arguing against the existence of God on the grounds that "I have not found Him in my test tube," or like the first Soviet cosmonauts' "argument" that they had found no God in outer space. Ex hypothesi, if God exists He is not found in a test tube or in space. That would make Him a chemical or a meteor. A taxi trip through Cleveland disproves quasars as well as a laboratory experiment disproves God, or brain chemistry dispro
761029
So you just need to show that for any x,y in A, f(x)=f(y) implies that x=y.  So let x,y be elements of A so that f(x)=f(y).  Since f(x)=f(y), and g is a function, then g(f(x))=g(f(y)).  Since gof is injective by assumption, this implies that x=y.  So we start with f(x)=f(y), and this implies x=y.  Therefore f is injective.  Cheers!
917328
Gopal is correct, except (a) he meant x =  -1should be excluded from the domain, and (b) x = 0 should also be excluded from the domain (since g(0) is undefined).
141458
Check out Godel's theorem.  The way I read it it states that given any language you can write sentences that are true, false, or neither true nor false (i.e. "This sentence is false.").  I see God as the language which is Truth.  If you have the concept of truth, falsehood is derivitive from truth.  So if you have a true omnipotent God who is all good, you can have the false concept of his opposite, suffering, or any other sort of bad or evil thing you can imagine.  Just because you can say it in the language of God, doesn't make it true.  The material world is an example of sentences which you can construct in the language of God which are not true, i.e. illusion or relative.  As our thought becomes more spiritual or absolute, we find that the so-called evil in the world falls away from our experience, not that we ignore it, but our thought changes the world within the sphere of our consiousness.  The world is our collective shared consiousness, lots of people think that it is all that is real and filled with both good and evil, so it sure seems like God must have created all this suffering, while in fact he is of purer eyes than to behold anything but pure 100% good everywhere.  Just because you can have the concept of suffering or anything else bad, doesn't make it necessarily real.  We are God's image and likeness, we have free will, therefore, we can believe that which is not true to be true, i.e. believe that illusion is reality.  Without free will, the universe would not be infinite.
10761
this is a common "proof" that is often bandied about, even though it's wrong.  \n\nLet a=b. \n\nmultiply by a:\na^2 = ab\n\nsubtract b^2:\na^2 - b^2 = ab - b^2\n\nfactor both sides:\n(a+b)(a-b) = (a-b)b\n\ncancel (a-b)\n(a+b) = b\n\nsubstitute a=b from the first step:\n2b = b\n\ndivide by b:\n2=1\n\nsubtract 1 from both sides\n1=0\n\n\nfortunately sanity is saved when you realize that the step where you cancel (b-a) from both sides is actually dividing by zero, since you specify beforehand that b=a.  this is not so much a proof as a mathematical fallacy.  see the link below for another, more complicated version of the same thing.