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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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Thezi
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    See here and here for some possible answers to your question. – Matthew Cassell Dec 13 '21 at 01:46
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    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. – JMoravitz Dec 13 '21 at 01:46
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    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: https://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) https://math.stackexchange.com/questions/2387286/ https://math.stackexchange.com/questions/1892181/ https://math.stackexchange.com/questions/260876/ https://math.stackexchange.com/questions/36289/ – Eric Nathan Stucky Dec 13 '21 at 01:50
  • $1+\infty$ 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. – Thomas Andrews Dec 13 '21 at 02:00
  • 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. – Bonnaduck Dec 13 '21 at 02:06
  • You are trying to intuit “infinity” without a definition, which any mathematician will tell you is a dangerous thing to do. It took until the 19th century to start discussing the infinite mathematically, and even then, it occasioned much backlash of a religious nature. Trying to do math without definitions (or stated axioms) is useless. – Thomas Andrews Dec 13 '21 at 02:07
  • Does this answer your question? Is infinity a number? – FShrike Dec 16 '21 at 14:05

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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.

Eric Nathan Stucky
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