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Ok so I'm arguing with this one person that says that $1 + \infty > \infty$ is true, and I disagree.

But I can't disprove their points.

My argument is that if $1 + \infty > \infty$ then there exists a number greater than $\infty$, disproving the concept of infinity, because you can't simply add $1$ to infinity, because infinity is ever increasing.

So new_infinity would just become "1 + infinity".

They argue that you can just substitute in $x$ for infinity and have the statement $1 + x > x$ which is true (but I don't think you can substitute a variable in for infinity).

I asked my math professor about this question and he said $1 + \infty > \infty$ is false, but I don't really remember the explanation.

Could someone explain it in layman's terms (and maybe i misheard my professor and it is true idk at this point)?

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    $\begingroup$ See here and here for some possible answers to your question. $\endgroup$ Dec 13, 2021 at 1:46
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    $\begingroup$ What do you mean by "greater than." Without an adequate and rigorous definition of that phrase, one which covers the infinite case, the question is meaningless. $\endgroup$
    – JMoravitz
    Dec 13, 2021 at 1:46
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    $\begingroup$ Hey Thezi: this is a great question, but it is also one that has been discussed endlessly on this site, so it might attract some negative attention. Here are some related questions that might interest you: math.stackexchange.com/questions/551123 (<< I think this is closest to what you are asking: replace their 10 and 1 with your 1 and 0) math.stackexchange.com/questions/2387286 math.stackexchange.com/questions/1892181 math.stackexchange.com/questions/260876 math.stackexchange.com/questions/36289 $\endgroup$ Dec 13, 2021 at 1:50
  • $\begingroup$ $1+\infty$ is undefined. When $\alpha$ is an infinite cardinal or infinite ordinal, $1+\alpha=\alpha.$ But in the ordinal case, $\alpha+1\neq \alpha.$ If $f(x)\to\infty$ as $x\to a,$ $1+f(x)\to\infty,$ too. There are other meanings of infinity, but $1+\infty$ is usually not defined. $\endgroup$ Dec 13, 2021 at 2:00
  • $\begingroup$ You can't substitute $\infty$ for the real variable $x$, since $\infty$ is not a number, so your friend's reasoning is wrong. However, your reasoning is wrong, since you say $1+\infty>\infty$ implies that a number is larger than $\infty$, but $1+\infty$ is not a number. $\endgroup$
    – Bonnaduck
    Dec 13, 2021 at 2:06

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The short reason that your debate partner's argument is invalid is that $1+x>x$ is only true in certain contexts— in other words, you can't just substitute anything for $x$ and expect it to be true, or even meaningful. Such subtleties don't usually bother us because, for instance, this inequality holds for all $x$ that come from an ordered field (such as the real numbers $\Bbb{R}$), but infinity as it is usually understood cannot exist in an ordered field.

However, there is some truth to what they are saying. For instance, the equation $x+1>x$ is true in ordinal arithmetic. (Amusingly, the equation $1+x>x$ is not true using the standard definition of $+$ for ordinals; such is the weirdness that arises when we try to make infinite things precise.)

Like many paradoxes in mathematics at this level, this one arises because we assume that we can use our informal understanding of objects (in this case, $+$, $>$, and $\infty$). Once one formally defines what one means, these problems tend to go away. Thus the question becomes what sort of definitions one should accept, but this is often not in the scope of mathematics.

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