πŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

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This air-quote #Mathematics unquote article https://www.quantamagazine.org/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science-20251121/ keeps appearing in my feed, and I initially made some comments the first time, debunking it from a #Maths point-of-view, but given how it keeps popping up I think I need to do a more thorough #MathsMonday thread about it

Firstly, the author is a Physics journo, so you can take what he says about #math with a grain of salt (for some reason I see a lot of them doing this overreach, instead of checking with a Mathematician)…

Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Dec 1

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"the strange infinite ones that other mathematicians ignore" - Mathematicians "ignore" them because there's no such thing

"Set theory deals with the infinite" - no it doesn't. Sets are finite, literally have an end. They have cardinality - the number of elements in the set. You can't have a cardinality of "infinity", nor is it a set if there's no closing bracket (which there can't be if it's infinite)…

Show threadπŸ‡ΊπŸ‡¦ haxadecimal πŸš«πŸ‘‘Feb 27
@SmartmanApps
I'm not a mathematician, but in multiple mathematics courses I've taken, set theory has been taught including both finite and infinite sets.
If you don't believe Wikipedia, Wolfram MathWorld also discussed both finite and infinite sets.
As you say, an infinite set cannot be defined by enumerating its elements. However, that is not the only way to define a set.
There exist, for instance, infinite sets of the natural numbers, integers, rationals, algebraic numbers, reals, etc.
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Feb 27

@brouhaha
"I'm not a mathematician" - I'm a Maths teacher.

"infinite sets. If you don't believe Wikipedia" - you won't find infinite sets in Maths textbooks. Stop looking at Wikipedia for Maths - it can't even get order of operations correct! πŸ˜‚

"infinite sets of the natural numbers" - they're not a set, they're just "the Natural numbers".

Show threadActive MouseFeb 27

@SmartmanApps @brouhaha just chiming in here since I just followed haxadecimal, but there are plenty of textbooks that mention infinite sets.

This even starts implicitly at an early age - here's an example of a textbook which, on page 46, says that the solution to an inequality is "the set of all real numbers x that make the inequality true." So the solution to x < 1 is the set of all real numbers less than 1, and there are certainly infinitely many of those!

https://www.scribd.com/document/536766882/389788543-Edexcel-Pure-Maths-Year-1

But pick your favourite textbook on Group Theory, say - Jordan & Jordan is a good one. On page 41 they give examples of groups (which they define as a certain type of set) are the complex numbers, real numbers, natural numbers and other infinite sets.

Of course, any textbook on set theory will include infinite sets :) E.g. Halmos, Jech, ...

But let us call these things something other than sets. What Cantor showed remains true, whatever you call them: there is a bijection between the natural numbers and the rational numbers, but not between the natural numbers and the real numbers. I assume you don't disagree with that... or do you?

Edexcel Pure Maths Year 1 | PDF

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Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±

@ActiveMouse @brouhaha
"plenty of textbooks that mention infinite sets" - there were textbooks that used Lennes' order of operations, but they were wrong and faded away

"the solution to x < 1 is the set of all real numbers less than 1" - no, just all real x<1, as per this worksheet I was teaching last week

"there are certainly infinitely many of those!" - which is exactly why it's not a set. πŸ™„ I had a student who used some wrong notations for intervals, but notably he never used set notation

Mar 2 at 6:46amWeb
Show threadπŸ’‘πš‚π—†π–Ίπ—‹π—π—†π–Ίπ—‡ π™°π—‰π—‰π—ŒπŸ“±Mar 2
@ActiveMouse @brouhaha
"What Cantor showed remains true" - it was never true, having ignored that numbers are discrete. 0.5(0) with infinite zeroes still takes up half of the interval 0-1 on the number-line. Having infinite decimal places doesn't magically make it infinitesimal in size. The only number which takes up zero space on the number line is zero itself.