How Can Improbability and Infinitesimal Probabilities Exist in Real Life Events?

[PeroK]

No. One must distinguish between a logical certainty and a probability of one. They are not the same.
Suppose a number is selected randomly on the line segment [0,1]. The probability that the number is irrational is 1 because the subset of irrational numbers has a probability measure of 1. The rational numbers are countable and the rational subset has a probability measure of 0. If the number selected turns out to be rational, the consequences are not that "the wheels come off".

For example, if I select $1/\pi$, which is irrational, that is okay. But, if I select $0.5$, which is rational, then that is also okay. Hmm?

How would you tell whether the number you selected "turned out to be" rational?

 

[PeroK]

No. One must distinguish between a logical certainty and a probability of one. They are not the same.
Suppose a number is selected randomly on the line segment [0,1]. The probability that the number is irrational is 1 because the subset of irrational numbers has a probability measure of 1. The rational numbers are countable and the rational subset has a probability measure of 0. If the number selected turns out to be rational, the consequences are not that "the wheels come off".

I found something here. The reply from Kevin Carlson.

https://math.stackexchange.com/ques...andom-natural-number-and-a-random-real-number

He concludes that there is no way to pick a random real in the way that you describe, for example. I agree with this. As soon as you say "select a real number at random" you are no longer talking about something that makes mathematical sense.

 

[Quasimodo]

How many times you are picking out a number from [0,1] is what matters! The size of the sample space, There the probability becomes 1, even for 1 trial is only 0.
But this is an unfriendly forum. I'll leave you to sort it out yourselves...

 

[FactChecker]

I found something here. The reply from Kevin Carlson.

https://math.stackexchange.com/ques...andom-natural-number-and-a-random-real-number

He concludes that there is no way to pick a random real in the way that you describe, for example. I agree with this. As soon as you say "select a real number at random" you are no longer talking about something that makes mathematical sense.

It is easy to define ways to make a random selection of a point on a line segment. The only problem is in recording the selection with perfect accuracy, but that is another subject.

 

[PeroK]

You can select from a countable subset of real numbers. To do that, you choose a whole number $n$ and map that to $r_n$ where $\{r_n \}$ is a countable subset of real numbers. Trivially, for example, you can choose from $\frac{1}{\pi}, \frac{1}{2\pi}, \dots$.

Moreover, you start by doing what I claim is impossible: selecting lengths from the set of (all) real numbers! You can't do that either. This process is not well defined:

Suppose one has two line segments of different lengths, randomly chosen.

 

[FactChecker]

You can select from a countable subset of real numbers. To do that, you choose a whole number $n$ and map that to $r_n$ where $\{r_n \}$ is a countable subset of real numbers. Trivially, for example, you can choose from $\frac{1}{\pi}, \frac{1}{2\pi}, \dots$.

Moreover, you start by doing what I claim is impossible: selecting lengths from the set of (all) real numbers! You can't do that either.

I can let nature provide segments of different lengths and just select a line segment. Nature has no predisposition to any particular unit system. When I independently provide the units to determine the length, there is a probability of 1 that the length will be irrational.

 

[PeroK]

I can let nature provide segments of different lengths and just select a line segment. Nature has no predisposition to any particular unit system. When I independently provide the units to determine the length, there is a probability of 1 that the length will be irrational.

If I ask you for a random real number, then your answer might be: the width of your desk in $cm$? Something like that?

But, beyond a certain accuracy the width of your desk is not well-defined. Where exactly does it start and end to a scale less than an atom? And, if the atoms are moving, then the width is changing with time. Not to mention any QM uncertainties.

PS I would say that measurements are one way to generate random numbers. But, all measurement outcomes must be one of a finite (possibly countable) set of numbers. You can't have an uncountable number of possible outcomes from a measurement.

 

[FactChecker]

If I ask you for a random real number, then your answer might be: the width of your desk in $cm$? Something like that?

But, beyond a certain accuracy the width of your desk is not well-defined. Where exactly does it start and end to a scale less than an atom? And, if the atoms are moving, then the width is changing with time. Not to mention any QM uncertainties.

I am leaving the problem of accuracy as a separate subject and assuming infinite accuracy in this "thought experiment". Regardless of the uncertainties and the method used, as long as the units are human-defined, independent of the line segment selected, the result will be irrational with a probability of 1. A naturally occurring length does not have any predisposition to the rational numbers in any independently-defined, human-created unit system. The probability that the length is irrational is 1. If you are claiming that there is a predisposition to rational lengths in a unit system that it has no knowledge of, then you need to prove that.

 

[PeroK]

I am leaving the problem of accuracy as a separate subject and assuming infinite accuracy in this "thought experiment". Regardless of the uncertainties and the method used, as long as the units are human-defined, independent of the line segment selected, the result will be irrational with a probability of 1. A naturally occurring length does not have any predisposition to the rational numbers in any independently-defined, human-created unit system. The probability that the length is irrational is 1. If you are claiming that there is a predisposition to rational lengths in a unit system that it has no knowledge of, then you need to prove that.

Okay, I'll accept that as a hypothesis.

But, the number of lengths we can define is countable. We can start with your desk, your piano etc. Even if the universe is infinite, there are only a countable number of atoms, so only a countable number of things that can ever exist and have a length.

Now, we list these objects that (hypothetically could ever exist in this universe) and their lengths are $L_1, L_2, \dots$.

And now, you are selecting your real number from this countable subset of the real numbers.

If you appeal to nature, you do not have an uncountable number of lengths to choose from.

 

[FactChecker]

Suppose I define a line segment, with no knowledge of the units of measurement. Suppose that another person independently defines the units of length measurement. Although we must eventually give up on getting a completely accurate length determination, the fact remains that the length in those units is irrational with a probability of 1. That is forced by the enormously larger quantity of irrational numbers. There is NO positive probability that the length is rational.

 

[PeroK]

Suppose I define a line segment, with no knowledge of the units of measurement. Suppose that another person independently defines the units of length measurement. Although we must eventually give up on getting a completely accurate length determination, the fact remains that the length in those units is irrational with a probability of 1. That is forced by the enormously larger quantity of irrational numbers. There is NO positive probability that the length is rational.

First, a line segment is a mathematical object. It cannot be physically measured.

Second, I'm not sure it's well defined to talk about "the probability that the length of an object is irrational". That depends on how you define length; which is a physical process. One process may define length one way and another process a different way. There's no mathematical axiom to define the length of an object in one specific way.

Third, the real numbers are - whatever you say - a difficult, abstract mathematical construction. You can't define or study the real numbers by an appeal to nature. You cannot prove the Archimedian property, say, by an appeal to nature and a thought experiment about length measurments. You cannot prove anything about the real numbers by an appeal to nature.

Fourth, that the rational numbers have measure zero is an entirely mathematical statement. You cannot by mathematics alone prove that every length is an irrational number (in any system of units). There is always at least one system of units where a given length is precisely $1$ unit.

 

[FactChecker]

You seem to imply that objects in nature are all in a specific set of precise positions that can only be specified using the rational numbers. IMHO, that is wrong in general and especially wrong at the quantum level.

I do not consider the irrational numbers to be any stranger than the rational numbers. They are just a different sequence of digits, but far less restricted and more numerous. The fact that humans want to define numbers using a finite series of arithmetic operations reflects a lack of imagination on the part of humans.

PS. I think this discussion has gotten too philisophical for me. I will leave further comments up to smarter (and wiser) people than myself.

 

[PeroK]

You seem to imply that objects in nature are all in a specific set of precise positions that can only be specified using the rational numbers.

I've no idea where you think I've said that. I said there are only a countable number of objects in nature.

I do not consider the irrational numbers to be any stranger than the rational numbers.

"Strange" is not a precise term. But, for example, most real numbers (all but a countable subset) are not computable.

The issue of non-computability is the real issue here. Not rationality/irrationality. And countability.

The fact that humans want to define numbers using a finite series of arithmetic operations reflects a lack of imagination on the part of humans.

Given that the real numbers are a human mathematical construction, I can't see the point of this statement.

It's been an interesting debate, but I'm not sure there is any point in taking it further.

 

[Twodogs]

"By default, mathematical reasoning is understood to take place in a deterministic mathematical universe. In such a universe, any given mathematical statement (that is to say, a sentence with no free variables) is either true or false, with no intermediate truth value available. Similarly, any deterministic variable can take on only one specific value at a time."

This makes sense, but there is always the question of whether the attribute of being deterministic is a projection of our mathematics upon a physical universe that is not so rigidly governed. The word determine connotes limitation, a meaning that can be traced back to the PIE root meaning "peg, post; boundary marker." Perhaps, in the physical universe, limitation has its limits.

As to making a random choice, there has been much discussion about whether this is possible in a deterministic universe. Can one actually build a device that makes a random choice without reference to a truly "random" physical process?

 

[sysprog]

Suppose I define a line segment, with no knowledge of the units of measurement. Suppose that another person independently defines the units of length measurement. Although we must eventually give up on getting a completely accurate length determination, the fact remains that the length in those units is irrational with a probability of 1. That is forced by the enormously larger quantity of irrational numbers. There is NO positive probability that the length is rational.

Saying that the probability that a number chosen at random will be irrational is one is logically equivalent to saying that the probability that it will be rational is zero. I reject this expedient as incorrect language. Only the impossible has probability zero, and only the absolutely certain has probability one. The probability that the number will be irrational is actually one minus the possibility or probability that it will be rational. Because it is not impossible that the number could be rational, the probability that it will be rational is not zero.

 

[PeroK]

Saying that the probability that a number chosen at random will be irrational is one is logically equivalent to saying that the probability that it will be rational is zero. I reject this expedient as incorrect language. Only the impossible has probability zero, and only the absolutely certain has probability one. The probability that the number will be irrational is actually one minus the possibility or probability that it will be rational. Because it is not impossible that the number could be rational, the probability that it will be rational is not zero.

The root of the problem is the incorrect application of mathematics (probability theory) to number selection (algortithms and computability). In abstract measure-probability theory:

If $X$ is a random variable distributed uniformly on $[0,1]$, then the following are true:

$\forall x: p(X = x) = 0$

$p(x \in \mathbb Q) = 0$

$p(x \notin \mathbb Q) = 1$

Note that, as pointed out above, abstract probability theory says nothing about "possible" or "impossible". That's an interpretation of the mathematics.

The real problem comes when one invokes this to say something like "if you choose a number in $[0,1]$, the probability it is rational is zero." Some people go further and say "whatever number you choose, the impossible has happened." If you look online, you will see this nonsensical assertion in many places.

This I believe to be misguided and a mis-application of abstract probability theory.

There is no algorithm that can select from more than a countable predefined set of numbers. Not least because the set of real numbers that you can even describe (the "computable" numbers) is countable.

This is why the impossible (something with a probability of $0$) can't "happen". There's no real-world algortithm or process to which the mathematics applies.

PS The probability that a number chosen at random is rational depends on the algorithm with which you choose the number. You can assign a probability to $p(x \in \mathbb Q)$ based on an analysis of your algortithm. It will be zero only if your algorithm is incapable of selecting any rational number! But not otherwise.

PPS all mathematics is physically "impossible" if you think about it. Something like "let $f(x) = \sin x$" conjures a physically impossible infinite sine function. Similarly, and rather prosaically actually, "let $X$ be a random variable uniformly distributed on $[0, 1]$" conjures a physical impossibility. It's just that in this case some people confuse themselves by imagining that what you can do mathematically with numbers you must be able to do physically with numbers.

 

[sysprog]

The root of the problem is the incorrect application of mathematics (probability theory) to number selection (algortithms and computability). In abstract measure-probability theory:

If $X$ is a random variable distributed uniformly on $[0, 1]$, then the following are true:

$\forall x \ p(X =x) = 0$

$p( x \in \mathbb Q) = 0$

$p( x \notin \mathbb Q) = 1$

Note that, as pointed out above, abstract probability theory says nothing about "possible" or "impossible". That's an interpretation of the mathematics.

The real problem comes when one invokes this to say something like "if you choose a number in $[0,1]$, the probability it is rational is zero." Some people go further and say "whatever number you choose, the impossible has happened." If you look online, you will see this nonsensical assertion in many places.

This I believe to be misguided and a mis-application of abstract probability theory.

There is no algorithm that can select from more than a countable predefined set of numbers. Not least because the set of real numbers that you can even describe (the "computable" numbers) is countable.

This is why the impossible (something with a probability of $0$) can't "happen". There's no real-world algortithm or process to which the mathematics applies.

PS The probability that a number chosen at random is rational depends on the algorithm with which you choose the number. You can assign a probability to $p(x \in \mathbb Q)$ based on an analysis of your algortithm. It will be zero only if your algorithm is incapable of selecting any rational number! But not otherwise.

A problem with that analysis is that: $\exists x \ (X =x)$ is asserted implicitly by the postulate that a number is selected, and that is provably inconsistent with $\forall x \ p(X =x) = 0$.

 

[PeroK]

A problem with that analysis is that: $\exists x \ (X =x)$ is asserted implicitly by the postulate that a number is selected, and that is provably inconsistent with $\forall x \ p(X =x) = 0$.

That's mathematical "selection". That's not something that you can "really do". If you reserve "possible/impossible" to describe something you can actually do, then the problem disappears.

You must see $p(X = x) = 0$ or $\mu(\mathbb Q) = 0$ as mathematical statements.

You can't argue on physical grounds that $\mu(\mathbb Q) \ne 0$ because "rational numbers exist". The measure is well-defined and the measure the rational numbers is 0.

Likewise, you mustn't take this to say "rational numbers are impossible to find".

 

[sysprog]

Again, I claim that saying that something which is obviously not impossible has a zero probability is a pet inconsistency in the use of language. It's logically provable that the impossible, and only the impossible, has zero probability. I disagree with LeBesgue's use of zero for the measure of the rationals. I regard it as a misuse of 'zero' and consequently as incorrect language. I would assign measure zero only to the empty set. I have no problem with assigning the rationals a measure $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0));$$ i.e. the measure is no less than some minimally positive number and therefore is non-zero.

 

[WWGD]

Again, I claim that saying that something which is obviously not impossible has a zero probability is a pet inconsistency in the use of language. It's logically provable that the impossible, and only the impossible, has zero probability. I disagree with LeBesgue's use of zero for the measure of the rationals. I regard it as a misuse of 'zero' and consequently as incorrect language. I would assign measure zero only to the empty set. I have no problem with assigning the rationals a measure $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0));$$ i.e. the measure is no less than some minimally positive number and therefore is non-zero.

But there is no "Minimally-positive" Standard Real number.

 

[sysprog]

I think any definition set which requires admission of such absurdities as the notion that the set of rational numbers has 'zero content' is faulty.

The cardinality of any non-empty set is strictly greater than that of the empty set. The cardinality of the rationals is strictly greater than that of any finite set. The cardinality of the reals is strictly greater than that of the rationals. The cardinality of the power set of the reals is strictly greater than that of the reals.

None of those remarks about cardinalities is linguistically self-inconsistent or inter-inconsistent. Please recall that my objection is to the misuse of language; not to the mathematical insights.

But there is no "Minimally-positive" Standard Real number.

That's an informal description of what was intended by the reference to $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0)).$$Another way to describe that is that it is a positive number $x$ such that any other positive number is either equal to $x$ or greater than $x$. I can't say the value of $x$ but I can indicate that it has that contemplated property and let that suffice because I can't do better. To call it zero would be inconsistent with calling it positive. Once you say a number is positive you can't consistently with that statement also say it is zero.

 

[WWGD]

I think any definition set which requires admission of such absurdities as the notion that the set of rational numbers has 'zero content' is faulty.

The cardinality of any non-empty set is strictly greater than that of the empty set. The cardinality of the rationals is strictly greater than that of any finite set. The cardinality of the reals is strictly greater than that of the rationals. The cardinality of the power set of the reals is strictly greater than that of the reals.

None of those remarks about cardinalities is linguistically self-inconsistent or inter-inconsistent. Please recall that my objection is to the misuse of language; not to the mathematical insights.

That's an informal description of what was intended by the reference to $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0)).$$Another way to describe that is that it is a positive number $x$ such that any other positive number is either equal to $x$ or greater than $x$. I can't say the value of $x$ but I can indicate that it has that contemplated property and let that suffice because I can't do better. To call it zero would be inconsistent with calling it positive. Once you say a number is positive you can't consistently with that statement also say it is zero.

It's the best we have thus far. Our probability theory on subsets of the Reals does not have enough resolution to distinguish impossible events outside of the sample space and sets with countably-many elements. How do we improve on this? I am not sure.

 

[sysprog]

It's the best we have thus far. Our probability theory on subsets of the Reals does not have enough resolution to distinguish impossible events outside of the sample space and sets with countably-many elements. How do we improve on this? I am not sure.

You just ably made the distinction in the very act of denying the ability to do so.

 

[WWGD]

You just ably made the distinction in the very act of denying the ability to do so.

I am not saying there is no distinction, just that our present Mathematical models don't allow for an effective way of making it. Edit: to the best of my knowledge.

 

[Stephen Tashi]

It's logically provable that the impossible, and only the impossible, has zero probability.

What is the proof? Logic can prove nothing by itself without assumptions or definitions.

I disagree with LeBesgue's use of zero for the measure of the rationals.

That's a statement of your personal preference. If you can propose a different probability measure then this can be discussed in the context of mathematical probability theory.

I have no problem with assigning the rationals a measure $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0));$$ i.e. the measure is no less than some minimally positive number and therefore is non-zero.

The fact a definition is made doesn't prove the thing defined actually exists. (It also doesn't prove the thing defined is unique - even if it does exist.) Further, defining a probability measure for a certain type of subset of [0,1] doesn't completely define the measure. It must be defined for all subsets of some sigma algebra of sets. As someone suggested in an earlier post, your might be able to implement the concept of a "minimally positive number" by extending the real number system as in done in non-standard analysis https://en.wikipedia.org/wiki/Non-standard_analysis. Perhaps somebody has already worked this out.

 

[FactChecker]

Probabilities are not necessarily tied to the human ability to devise a finite, terminating selection process. There is much that happens and exists in nature that has probabilities with no human involvement and no known finite "selection" process.
Many comments in this thread are attempting to discard a great deal of standard probability theory that is based on measure theory. That would require a lot of work and would greatly increase the complexity of the theory. I am not sure that anyone here has identified a single benefit of that approach.

 

[sysprog]

What is the proof? Logic can prove nothing by itself without assumptions or definitions.

If a definition of a term can be shown to be inconsistent with another definition of the same term, that is in my view adequate proof of invalidity of at least one of the definitions.

That's a statement of your personal preference. If you can propose a different probability measure then this can be discussed in the context of mathematical probability theory.

I just did. This is a proposed non-zero definition whereby $x$ is the measure of the rationals: $$x ~|~[\{ x \in \mathbb R~ |~ x>0 \} \wedge \forall(y)[\{y \in {\mathbb R~ |~ y>0}\} \Rightarrow (y \geq x)]]~(\perp (x \neq 0)).$$ I understand that $x$ would thereby be small enough that it could be treated as zero, but it wouldn't thereby be asserted to actually be equal to zero.

The fact a definition is made doesn't prove the thing defined actually exists.

An inconsistent pair of definitions proves that at least one or the other does not actually exist.

(It also doesn't prove the thing defined is unique - even if it does exist.)

That's not something I'm quibbling about.

Further, defining a probability measure for a certain type of subset of [0,1] doesn't completely define the measure.

I'm objecting to inconsistency; not offering completeness.

It must be defined for all subsets of some sigma algebra of sets.

In my view, resorting to inconsistent definitions of zero to achieve this, while it is certainly convenient, is incorrect use of language, and therefore objectionable.

As someone suggested in an earlier post, your might be able to implement the concept of a "minimally positive number" by extending the real number system as in done in non-standard analysis https://en.wikipedia.org/wiki/Non-standard_analysis. Perhaps somebody has already worked this out.

I don't think that eliminating inconsistency in the use of the term 'zero' requires extending the reals beyond whatever is entailed by inclusion of the infinitesimal within the standard. It may require use of a different symbol, such as $0^+$, and a corresponding definition and set of rules, that allows an infinitesimal to be treated as zero without it being asserted to actually be zero.

 

[sysprog]

Probabilities are not necessarily tied to the human ability to devise a finite, terminating selection process. There is much that happens and exists in nature that has probabilities with no human involvement and no known finite "selection" process.
Many comments in this thread are attempting to discard a great deal of standard probability theory that is based on measure theory. That would require a lot of work and would greatly increase the complexity of the theory. I am not sure that anyone here has identified a single benefit of that approach.

I'm merely trying to object steadfastly to the complacent use of inconsistent definitions for zero.

 

[WWGD]

I'm merely trying to object steadfastly to the complacent use of inconsistent definitions for zero.

Please point out the inconsistency that follows from using zero as you mention. I don't see it.

 

[sysprog]

Please point out the inconsistency that follows from using zero as you mention. I don't see it.

Only the impossible actually has probability zero. To say of an event that it is possible for it to occur is to say that its probability of occurring, however small, is non-zero. Saying that if a positive quantity is so small that we can't measure it then it is equal to zero, is saying that the quantity is at once positive and therefore non-zero and also equal to zero and therefore non-positive. Nothing can be both zero and positive because the definition of positivity is that the referent is strictly greater than zero and therefore strictly not equal to zero.

 

[WWGD]

Only the impossible actually has probability zero. To say of an event that it is possible for it to occur is to say that its probability of occurring, however small, is non-zero. Saying that if a positive quantity is so small that we can't measure it then it is equal to zero, is saying that the quantity is at once positive and therefore non-zero and also equal to zero and therefore non-positive. Nothing can be both zero and positive because the definition of positivity is that the referent is strictly greater than zero and therefore strictly not equal to zero.

It seems you can just state or describe some elements, here countable subsets as being in the sample space yet with measure zero and others as not being in the sample space and avoid any confusion.

 

[sysprog]

It seems you can just state or describe some elements, here countable subsets as being in the sample space yet with measure zero and others as not being in the sample space and avoid any confusion.

I'm not trying to assert that the misuse of language to which I refer entails the existence of any confusion on the parts of those who so misuse language. I'm merely asserting that it's incorrect. Part of the definition of 'measure zero' is 'having zero content'. Saying that the set of rationals has 'zero content' is saying something that is patently false. Among sets and their subsets, only the empty set has zero content. That's what 'empty' means. Non-empty sets are non-empty because they have more than zero content. Again, I'm advocating for consistent use of language; not trying to cast aspersions on anyone's mathematical insights.

 

[PeroK]

Only the impossible actually has probability zero.

That depends on how you define "impossible". "Impossible", in mathematics usually has nothing to do with probabilities. Usually it means that some set is the empty set:

It's impossible to find a real solution to the equation $x^2 + 1 = 0$ is an informal way of saying: $\{ x \in \mathbb R : x^2 + 1 = 0 \} = \emptyset$.

It doesn't mean:

If you choose a real number, $x$, on a uniform distribution of all real numbers, then the probability that $x^2 + 1 = 0$ is zero.

Also, $\mu(\emptyset) = 0$, which means that the empty set has "probability" zero. But, that doesn't mean that the empty set is "impossible".

Your whole argument is based on a confusion of terminology.

 

[sysprog]

That depends on how you define "impossible". "Impossible", in mathematics usually has nothing to do with probabilities. Usually it means that some set is the empty set:

It's impossible to find a real solution to the equation $x^2 + 1 = 0$ is an informal way of saying: $\{ x \in \mathbb R : x^2 + 1 = 0 \} = \emptyset$.

It doesn't mean:

If you choose a real number, $x$, on a uniform distribution of all real numbers, then the probability that $x^2 + 1 = 0$ is zero.

It means that, too, because that too is entailed by the premises.

Also, $\mu(\emptyset) = 0$, which means that the empty set has "probability" zero. But, that doesn't mean that the empty set is "impossible".

I wouldn't say that the empty set is impossible, but I would say that the set of possible impossibilities is empty.

Your whole argument is based on a confusion of terminology.

I'm not the one who is confusing the terminology. It's inconsistent usage that confuses the terminology.

If you choose a real number, $x$, on a uniform distribution of all real numbers, then the probability that $x^2 + 1 = 0$ is zero.

That's fine. You have to resort to complex numbers to satisfy that equation. What I'm objecting to is, e.g., given a choice of a real number $x$ as you postulated, the assertion that the probability that $x - 1 = 0$ is zero. It's possible that $x=1$ because the specified conditions don't rule it out; wherefore, it has probability greater than zero.

 

[PeroK]

What I'm objecting to is, e.g., given a choice of a real number $x$ as you postulated, the assertion that the probability that $x - 1 = 0$ is zero. It's possible that $x=1$ because the specified conditions don't rule it out; wherefore, it has probability greater than zero.

Whether $p(1) = 0$ or not depends on the distribution. If the distribution is uniform on $[0,1]$, then $p(1) = 0$. I'm sure you know the argument.

This is all mathematics. There is no sense in which we are dealing with "possible" or "impossible" events. If you define these terms mathematically, then they have the properties they have through their definition. They do not have properties based on the English-language definition of the word used. If you define an "impossible" set as one having measure zero, then that is your definition. You can't invoke an English-language meaning of the word to override your mathematics.

Here is an example of where you are going wrong. One could argue that all numbers are "rational" because they all obey logic. One could argue that an "irrational" number is a contradiction. But, that argument confuses "irrational" as an English word; and "irrational" as a well-defined mathematical term.

You are likewise confusing "impossible" as an English word with a defintion inside probability theory. In one sense it's worse because actually "impossible" has no meaning inside probability theory, except as an informal term for a set of measure zero.

Another example:

All functions and matrices are "invertible" because you can write them upside down.