Prove These Two Propositions Check Out Description For Detail

Are there any mathematical propositions that were checked for billions of values all correct but later proved false?

It’s not clear what you’re asking. When a theorem is proved, it is proved for all values, not just for some. Once proved true, it can’t be proved false.But maybe you’re asking for examples of statements which are true for many values, but not for all. There are questions like that already on Quora. Here’s one that I’ve recently seen:

In a proof by contradiction, what if both the proposition and its negation lead to contradictions?

This is the business of logic, not mathematics per se.A formulation is not automatically a STATEMENT of logic. Only those those that have TRUE or FALSE as answer are admitted as logical statements (also called propositions). For example, the sentence “the smell of jack fruit is repulsive” is an opinion (actually expressed by some people!) and cannot be meaningfully decided as TRUE or FALSE, and not admitted as a proposition.The sentence “Sesame oil can be beaten to any flat shape at room temperature” is admissible (and happens to be FALSE).Now logic there is a principle called THE LAW OF EXCLUDED MIDDLE which states that a statement is either TRUE or FALSE and cannot be anything in between.So when a valid proposition, if it it leads to a contradiction is to be interpreted as the negation to be TRUE.If the negation leads to contradiction mostly it means some poor formulation, possibly it is not a logical statement or the “negation” is what one thinks it is.EXAMPLE: “Two persons can together can lift a car”If you change “two” to “one” in the above that is not the negation.

Proof Verification (Analysis): I have written a proof of a proposition concerning subsequences. Can you please check whether it is correct or not?

Two thingsVery clever writing the proof in the comments. I had this problem with "long" comments in Quora too. The proof is very well written, by which I mean that it is formatted very well and is clear to read. Coming to whether the proof is correct, it is technically correct. However, instead of using [math]M=f(N)[/math], [math]M=N[/math] will suffice. A subsequence can be looked at as a sequence in which you knock out infinitely many terms, but still leave an infinite number of terms behind, meaning it is still a sequence. You are not allowed to rearrange anything. When you say that for the sequence [math](a_n)^{n=\infty}_{n=0}[/math], given an [math]\epsilon[/math], there is an [math]N[/math] in natural numbers, you are saying you that any term from the [math]N^{th}[/math] term works. In a subsequence [math](b_n), a_N=b_N[/math] only if you knocked out no terms till then. Otherwise, it would appear sooner meaning that in most cases, [math]N[/math] is a safer choice in the subsequence than it is in the original sequence itself. Using [math]f(N)[/math] is overkill, thought not wrong. I would suggest you take a sequence, build a subsequence and convince yourself of it.

I have written a proof of a proposition about the prime numbers. Can you please check whether my proof is correct or not? (see the question details)

The proof is essentially correct except for the part you claim it must converge. You need to use full sandwich theorem both upper and lower bound converges to same value 0 and 1 are useless here. Instead note that it is monotonous because each multiplication decreases the value and is bounded under. This is called monotone convergence theorem.

Is the statement "God Exists" a synthetic or analytical proposition?

Is the statement "God Exists" a synthetic or analytical proposition?If such a statement is a synthetic proposition, then we would need experimental evidence to prove it. That would be without regard to any logical argument for the existence of God. Do believers in God recognize proving God logically is pointless and we would need experimental evidence to prove God.Careful analysis leads to the conclusion that written down in this form it is neither, it is an ambiguous proposition.AnalyticalIn an analytic proposition we only need the exact meaning of the words to conclude whether it is true.Pigs are animals, is an analytical proposition. When you know what the words mean you know it is true.SyntheticIn a synthetic proposition we need to combine knowledge of the meaning of the words with sufficient relevant knowledge about the world to conclude whether it is true.Not all pigs are dirty, is a synthetic proposition. We need to know the words and check the world. Because we have a clear unambiguous meaning of the words clean and pig. We know what real pigs are like, we know what we look for and as soon as we find a clean pig we know it is true.AmbiguousWe do not know very well what the word God means, because there is no such thing as a universal agreement what a real God is. This means that as a proposition, it is ambiguous.The ontological argument uses a God definition that tries to makes the proposition analytical.Other type of argument use a God definition that makes the proposition synthetic, such as the cosmological argument, the teleological argument, the moral argument and arguments from religious experience and miraclesThe proposition "God exists" is in itself to ambiguous to be either of the two.

How do we know that proof by induction actually proves what we are trying to show?

Dan Christensen has already given a good answer, and I can only illustrate it by giving an example from the classical textbook “Foundation of Analysis” by Edmund Landau. He clearly states the axioms for natural numbers:His book is famous for developing everything from first principles, and the Axiom of Induction is one of them. In his “Preface for the tudent” he emphaises that he starts from scratch:And further:As you can see, in proper development of arithmetic, Axiom of Induction preceeds [math]2 \times 2 = 4[/math] .