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Although the other answers correctly explain the basic logical equivalence of the two proof methods, I believe an important point has been missed: With good reason, we mathematicians prefer a direct …

The question becomes interesting when it is interpreted as a technical question about the extent to which we can have a semi-formal language somehow in-between the truly formal proofs, which are large …

Logicians are familiar with the variety of self-referential sentences expressible in the language of arithmetic: The Gödel sentence, "this sentence is not provable", which indeed is not provable in w …

My example is the classical proof that sqrt(2) is irrational. More generally, many proofs that proceed by showing that there are no minimal counterexamples exemplify your phenomenon. The method of no …

Set theory provides a good example. It is often convenient in set theory to work with the concept of "classes" and treat them as mathematical objects of their own kind. The standard axiomatization of …

Although your question is vague in certain ways, one robust answer to it is provided by the subject known as Reverse Mathematics. The nature of this answer is different from what you had suggested or …

You must read the charming essay lampooning this notation, while also giving a thorough logical analysis of it, by Adrian Mathias. Adrian Mathias, A Term of Length 4,523,659,424,929, Synthese 133 (20 …

The claim you have asked us to prove is not true. If PA is consistent, then by the Incompleteness Theorem there are $\Pi_1$ statements that are independent of PA, such as Con(PA), which can be seen to …

Surprisingly, the answer is yes! Well, let me say that the answer is yes for what I find to be a reasonable way to understand what you've asked. Specifically, what I claim is that if PA is consistent …

There are indeed many proofs of the Compactness theorem. As I mention in this MO answer, when I was a graduate student Leo Harrington told me that he used a different proof method for Compactness eac …

There is a body of very interesting work surrounding the proof complexity of various formulations of the well-known pigeon-hole principle, the fact that there is no injective function from a set of si …

With regard to your sub-question, Now imagine a universe where $\text{Con}(\text{ZFC})$ holds but all the models of $\text{ZFC}$ are $\omega$-nonstandard and believe $\neg \text{Con}(\text{Z …

The Truth Lemma The result says that what's true in a forcing extension $M[G]$ is just what's forced to be true by the path of the generic filter $G$. More precisely: Suppose $M$ is a countable t …

$\newcommand\Con{\text{Con}} \newcommand\Dec{\text{Dec}}$ Let $F$ be the formal system in which the proofs are to be carried out, when it comes to your formal assertions of the form $\Dec(\varphi)$. …

The answer is no. Choose a fundamental sequence for $\epsilon_0$ itself in the usual way, which I think is $\epsilon_0[n]=\omega^{\omega^{{\vdots}^\omega}}$, and then modify the earlier fundamental se …