Question Number 205577 by BaliramKumar last updated on 25/Mar/24
$$\mathrm{is}\:\infty\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}? \\ $$
Commented by mr W last updated on 25/Mar/24
$${it}\:{is}\:{only}\:{a}\:{symbol}.\:{it}\:{may}\:{have}\:\: \\ $$$${different}\:{meanings}\:{depending}\:{on}\: \\ $$$${in}\:{which}\:{context}\:{it}\:{is}\:{used}.\: \\ $$$${generally}\:{people}\:{use}\:{it}\:{to} \\ $$$${describe}\:{something}\:{without}\:{any}\: \\ $$$${limit}\:\left({boundary}\right). \\ $$
Commented by BaliramKumar last updated on 25/Mar/24
Commented by mr W last updated on 25/Mar/24
$${here}\:{a}\:{context}\:{is}\:{given},\:{the}\:{topic}\:{is} \\ $$$${about}\:{numbers}.\:{when}\:{talking}\:{about} \\ $$$${numbers},\:\infty\:{refers}\:{to}\: \\ $$$$−\infty<{x}<+\infty \\ $$$${but}\:{still}\:{we}\:{can}\:{not}\:{say}\:\infty\:{is}\:{a}\:{number} \\ $$$${or}\:\infty\:{is}\:{a}\:{real}\:{number}. \\ $$
Commented by BaliramKumar last updated on 25/Mar/24
$$\mathrm{i}\:\mathrm{think}\:\mathrm{Q}\:\mathrm{is}\:\mathrm{wrong} \\ $$$$\mathrm{e}\:\mathrm{is}\:\mathrm{also}\:\mathrm{irrational} \\ $$
Commented by mr W last updated on 25/Mar/24
$${yes},\:{question}\:{is}\:{not}\:{a}\:{good}\:{one}.\:{the}\: \\ $$$${answer}\:{given}\:{is}\:{also}\:{wrong}. \\ $$
Answered by TheHoneyCat last updated on 31/Mar/24
![In standard math (Russel′s set theory, Bourbaki′s construction... ect...) R (the set of real numbers) is defined as the completion of Q (i.e. the limits of bounded increasing sequences in Q). So in all the usual meanings of “∞” ∞∉R This includes: formal topological closure of R R^ =R⊎{+∞,−∞} The projective plane RP^1 where ∞:=[0:1] The sur−real extension ∞=ℵ_0 (or any “infinite” number) But of course, there are numerous areas of math where this is not explicitely stated and we don′t pay attention wether R refers to real numbers or some extentions...](Q205851.png)
$$\mathrm{In}\:\mathrm{standard}\:\mathrm{math}\:\left(\mathrm{Russel}'\mathrm{s}\:\mathrm{set}\:\mathrm{theory},\right. \\ $$$$\left.\mathrm{Bourbaki}'\mathrm{s}\:\mathrm{construction}...\:\mathrm{ect}...\right) \\ $$$$\mathbb{R}\:\left(\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{real}\:\mathrm{numbers}\right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{the} \\ $$$$\mathrm{completion}\:\mathrm{of}\:\mathbb{Q}\:\left(\mathrm{i}.\mathrm{e}.\:\mathrm{the}\:\mathrm{limits}\:\mathrm{of}\right. \\ $$$$\left.\mathrm{bounded}\:\mathrm{increasing}\:\mathrm{sequences}\:\mathrm{in}\:\mathbb{Q}\right). \\ $$$$\mathrm{So}\:\mathrm{in}\:\mathrm{all}\:\mathrm{the}\:\mathrm{usual}\:\mathrm{meanings}\:\mathrm{of}\:``\infty'' \\ $$$$\infty\notin\mathbb{R} \\ $$$$\mathrm{This}\:\mathrm{includes}:\:\mathrm{formal}\:\mathrm{topological}\:\mathrm{closure}\:\mathrm{of}\:\mathbb{R} \\ $$$$\bar {\mathbb{R}}=\mathbb{R}\biguplus\left\{+\infty,−\infty\right\} \\ $$$$\mathrm{The}\:\mathrm{projective}\:\mathrm{plane}\:\mathbb{RP}^{\mathrm{1}} \:\mathrm{where}\:\infty:=\left[\mathrm{0}:\mathrm{1}\right] \\ $$$$\mathrm{The}\:\mathrm{sur}−\mathrm{real}\:\mathrm{extension}\:\infty=\aleph_{\mathrm{0}} \:\left(\mathrm{or}\:\mathrm{any}\right. \\ $$$$\left.``\mathrm{infinite}''\:\mathrm{number}\right) \\ $$$$ \\ $$$$\mathrm{But}\:\mathrm{of}\:\mathrm{course},\:\mathrm{there}\:\mathrm{are}\:\mathrm{numerous}\:\mathrm{areas}\:\mathrm{of} \\ $$$$\mathrm{math}\:\mathrm{where}\:\mathrm{this}\:\mathrm{is}\:\mathrm{not}\:\mathrm{explicitely}\:\mathrm{stated} \\ $$$$\mathrm{and}\:\mathrm{we}\:\mathrm{don}'\mathrm{t}\:\mathrm{pay}\:\mathrm{attention}\:\mathrm{wether}\:\mathbb{R}\:\mathrm{refers} \\ $$$$\mathrm{to}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{or}\:\mathrm{some}\:\mathrm{extentions}...\: \\ $$