Blog4Tzy

Contents    Proposition    Remark 1    Proof Proposition Let m 0 be a natural number. And let P ( m ) be a property pertaining to an arbitrary natural number m. Suppose that for each m ≥ m 0 , we have the following implication: if P ( m ′ ) is true for all natural numbers m 0 ≤ m ′ < m , then P ( m ) is also true. (In particular, this means that P ( m 0 ) is true, since in this case the hypothesis is vacuous.) Then we can conclude that P ( m ) is true for all natural numbers m ≥ m 0 . Remark 1 Since the principle of induction closes with a universal conclusion that the given property is true for all natural number x, we would intuitively apply it to the strong principle of induction because in the cases where x < m 0 , P ( x ) would be vacuously true. Whe...

This blog was previously held in the same URL (www.errortang.eu.org, TZY's land of outlaw) but the server was built by myself using Node.js and ngrok. However, after BOTH of my ngork service providers were under attack and no longer able to provide stable services, I have realized the cost of freedom of speech is yet too high to afford. My server was shut down for months and I rethought about why I need a website in the first place. The main reason is that I want my writings to be reviewed, discussed, commented and criticized publicly or privately by academic peers. Peer-review and academic community is probably the most important part of today's academic tradition, but it is also far from satisfying. For example, discussions and criticism are invaluable to a scholar, but sometimes you will hear more vacuous odes and hymns than constructive criticism on conferences, book reviews and sessions. I believe people do that partly because they are trying to network, and partly because...