Require Import Utf8_core.Īxiom A2: ∀ p q r : Prop, (p → q → r) → (p → q) → (p → r).Īxiom A3: ∀ p q : Prop, (¬ q → ¬ p) → (¬ q → p) → q. This kind of proof has to be done in a proof assistant to be believed. Here's an example: proving that implication is transitive in Hilbert's axiom system for propositional calculus. I'm pretty sure by arcane formalism OP means results in logic where the formalism is the entire difficulty of the proof. More originality than most of us possess to prove this lemma unless we were given Time and effort, of course and one would not expect the student to handle the trickierĮxamples.) But the Urysohn lemma is on a different level. Through the book and work out the proofs independently. To a bright student who had not studied topology, that student ought to be able to go What we mean this way: By and large, one would expect that if one went through thisīook and deleted all the proofs we have given up to now and then handed the book Why do we call the Urysohn lemma a “deep” theorem? Because its proof involvesĪ really original idea, which the previous proofs did not. Here is Munkres introducing a hard theorem in chapter 4 of 'Topology' The three theorems are (1) the intermediate value theorem, and a continuous function on a closed interval is (2) bounded and (3) achieves its supremum. In this context "hard" means making essential use of the completeness property not nearly as hard as the prime number theorem but a major step up in sophistication for the intended reader. TeX all the things Chrome extension (configure inline math to use delimiters) MathJax userscript (userscripts need Greasemonkey, Tampermonkey or similar) To view LaTeX on reddit, install one of the following: If you feel you were banned unjustly, or that the circumstances of your ban no longer apply, see our ban appeal process here.Ĭareer and Education Questions - every ThursdayĪ Compilation of Free, Online Math Resources. If you post or comment something breaking the rules, the content may be removed - repeated removal violations may escalate to a ban, but not without some kind of prior warning see here for our policy on warnings and bans. This subreddit is actively moderated to maintain the standards outlined above as such, posts and comments are often removed and redirected to a more appropriate location. Unnecessarily combative or unkind comments may result in an immediate ban. racism, sexism, homophobia, hate speech, etc.). This includes not only comments directed at users of /r/math, but at any person or group of people (e.g. If you upload an image or video, you must explain why it is relevant by posting a comment providing additional information that prompts discussion.ĭo not troll, insult, antagonize, or otherwise harass. Memes and similar content are not permitted. Image/Video posts should be on-topic and should promote discussion. If you are asking for advice on choosing classes or career prospects, please post in the stickied Career & Education Questions thread. Rule 4: No career or education related questions If you ask for help cheating, you will be banned. Do not ask or answer this type of question in /r/math. Homework problems, practice problems, and similar questions should be directed to /r/learnmath, /r/homeworkhelp or /r/cheatatmathhomework. This includes reference requests - also see our list of free online resources and recommended books. If you're asking for help learning/understanding something mathematical, post in the Quick Questions thread or /r/learnmath. Requests for calculation or estimation of real-world problems and values are best suited for the Quick Questions thread, /r/askmath or /r/theydidthemath. For example, if you think your question can be answered quickly, you should instead post it in the Quick Questions thread. Questions on /r/math should spark discussion. Rule 2: Questions should spark discussion Please avoid derailing such discussions into general political discussion, and report any comments that do so. In particular, any political discussion on /r/math should be directly related to mathematics - all threads and comments should be about concrete events and how they affect mathematics. All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics.
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