3 ms·
A function has a fixed point when for some value of x, x = f(x). This is interesting because it's some change, that doesn't change the result. in the lingua fr
by vorotato 10y ago
A function has a fixed point when for some value of x, x = f(x). This is interesting because it's some change, that doesn't change the result.
in the lingua franca of javascript.
var timesThree = (x => x * 3)
//for this function 0 is a fixed point...
var a = 0
console.log(a == timesThree(a))
//7 is not a fixed point
var b = 7
console.log(b == timesThree(b))
- hackernewsacct 10y agoThanks for the response. My question is how does fixed point relate to the incomplete theorems and decidability?
- darkinvisible 10y agoThe key to e.g. Goedel's results is the Fixed Point Lemma: if ψ is a formula with v free then there is a sentence φ such that, provably, φ <=> ψ(<φ>) [where <..> is the numerical code of .. ]. The proof, if you are interested, is not difficult. Let sub be the function that describes, via codes, substituting the (numeral n_ of the number) n for a free variable v of a formula χ : sub(<χv>, n_) = <χn_>. Then the PROOF is: consider ψ(sub(v,v)), call it θv, let m be <θv> and let φ be θm_. Then, provably, φ <=> ψ(sub(m_,m_)) <=> ψ(sub(<θv>,m_)) <=> ψ(<θm_>) <=> ψ(<φ>). Ta-da! Goedel used this to get the "formula that says I am not provable", φ <=> not Prov(<φ>).