Elsewhere arose a discussion about logical clauses that can be made from indeterminate forms, in this case, namely $0 / 0$. Since $0 / 0$ is indeterminate form, can we make these logical clauses:
- $0 / 0 = 1$ is false?
- $0 / 0 \neq 1$ is true?
Or in more general form, if $x$ is not just undefined, but indeterminate form, and $y$ is defined and $y \in \mathbb{R}$, can we say that:
- $x = y$ is false?
- $x \neq y$ is true?
My thinking goes, that since $x$ cannot be determined in any way, first one is intuitively correct: any clause that tries to say that $x$ is something must be false. Based on simple logics, it follows that also all clauses that $x$ is something else than something that is defined, are true.
Counterargument is, that if $x$ is indeterminate form, we cannot say that it is not $y$, for that it would make a clause that $x$ is something, because it is at least not $y$, which cannot be true if $x$ is truly indeterminate (i.e. if $x$ is indeterminate, we cannot say that it isn't $y$. To my thinking, this leads to also back to that $x = y$ is at least not true. But with the same argumentation, it cannot be false either, and hence both logical clauses are themselves undefined in answer, not true but not false either.
And counter-counterargument is that, if we make a clause that says $x \neq y$, it does not make $x$ any less indeterminate, because it only says that $x$ is at least not that particular defined form, but still leaves open possibilities that $x$ is something else than $y$ or still completely indeterminate.
My thinking goes that if $x$ is indeterminate form, it means that $x \notin \mathbb{R}$ (or $\mathbb{R}$, for that matter), and hence all clauses that $x \neq y \in \mathbb{R}$, are true, because what we can say about indeterminate forms is that they are not in real number space. But that might be untrue as well, if it is also indeterminate whether indeterminate forms exist in real numbers or not.
What comes to undefined numbers, e.g. if instead of $0 / 0$ we talk about $a / 0$, where $a \neq 0$, it is more clear that they do not exist in real numbers, so if $x$ was only undefined, I believe $x \neq y$ is without doubt true. I.e. clauses that undefined numbers are always not equal to any or all of defined numbers are necessarily true, because it is basically the same clause that $x \notin \mathbb{R}$, which is true if $x$ is undefined.
The answer to this is probably very much not indeterminate, but with my basic knowledge of algebra a definitely correctly-argumented answer cannot be made.
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