Why is the notion of infinity useful?

Question

I quote from the book Number, The language of science of Tobias Dantzig

"Had we avoided,as Kronecker urged us to avoid, the introduction of infinite processes and consequently that of irrationals, the complex number would be just a pair of rational numbers and whatever reality or unreality we could ascribe to the rational would also reside in the complex But in the search for a field in which any equation of algebra would have a process, we were compelled to legitimize the infinite process, and the so called real number was the result"

And immediately after " the act of becoming invokes the infinite as the generating principle for any number"

Could someone expain these sentences?

What exactly is not clear? Have you seen this question and similar ones? — Dietrich Burde, Apr 26 '19 at 13:14

J.G. · Accepted Answer · 2019-04-26T13:27:49.157

3

Let's say your friend only believes in rational numbers, and you want to convince them other numbers are worth using. Your friend used to only believe in integers, but accepted rational numbers to solve equations such as $2x-1=0$. This inspires you to point to polynomials over $\Bbb Q$ that $\Bbb Q$ cannot solve. Historically, $x^2-2=0$ (which would help you motivate irrational real numbers) sprang to mind before $x^2+1=0$ (which would help you motivate complex numbers).

But arguing from polynomials only motivates a countably infinite number system, such as $\Bbb A$ rather than the full $\Bbb C$. Getting to uncountably infinite number systems such as $\Bbb R$ or $\Bbb C$ requires other tricks (you have several options). These are the kinds of "infinite process" Dantzig has in mind.

edited Apr 26 '19 at 13:27

answered Apr 26 '19 at 13:22

J.G.

118,053

+1 but you meant to say "uncountably infinite number systems such as $\Bbb{R}...$ – saulspatz Apr 26 '19 at 13:26
@saulspatz Thanks; fixed. – J.G. Apr 26 '19 at 13:28
Thank you very much – veronika Apr 26 '19 at 16:19

Why is the notion of infinity useful?

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