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I originally asked this question on Yahoo! answers close to a decade ago, with a disappointing response, so I figured I'd try to post it here with hopefully better results.

Two engineers arguing over ridiculous semantics... the conversation goes like this:

"The feature set is infinitely better than the original."

"It can't be infinitely better, it can only be finitely better."

"The original feature set didn't have this feature, therefore it's infinitely better."

"That's better in an undefined, unquantitative sort of way; not 'infinitely'. At best it's finitely better, from a zero state to a one state."

"I disagree."

"Well you're a poopyhead."

"Fartknocker."

..and so the debate degenerated.

I'm hoping a mathematical purist can answer the question before we both end up as poster children for "Why engineers shouldn't breed."

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    this seems for plilosophical than mathematical – 2017-01-19
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    Some nonzero real numbers are less than zero. In the complex numbers there is no systematic ordering that tells which values are "more than zero". You are asking for "mathematical terms" but fail to define them, which leads to the primarily opinion based conversation described in this Question. – 2017-01-19

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Honestly this comes down to whether you measure improvements by subtraction or division.

More specifically: If some value increases from $15$ to $45$, you could say any of the following:

  • "It increased by $30$"
  • "It increased by a factor of $3$"
  • "It increased by $200\%$"

All three of those are accurate. The first one describes the change in absolute terms; the other two both describe the change in relative terms, but the middle expression compares final to initial, whereas the last expression compares change to initial.

If the initial value is $0$, though, things get complicated. Going from $0$ to $1$ can be called an increase of $1$ (additive description), but it cannot easily be described using a multiplicative description. It would certainly not be correct to say that it is "Infinitely more", but I suppose you could say that going from $0$ to $1$ is "infinitely many times more". That description is fairly useless, though, because there is no way to distinguish that change from any other change from $0$ to a positive number.

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In mathematical terms, a nonzero number can not be compared this way with zero. A sentence like "$1$ is ... times bigger than $0$" simply does not make any sense. Even if you fill the ellipsis with the words "infinitely many".