Does a photon not having a reference frame result in logical inconsistency?

Question

So, I know that SR makes two statements. One is the first assumption it has:
"Speed of light is the same with respect to all reference frames".

Second, the statement:
"A photon cannot have a reference frame".
(Please correct me if the statements I wrote are not accurate.)

Aren't these two statements a bit paradoxical?
One first says that Speed of this thing is the same with respect to all frames. And then says speed is not defined wrt to it's own frame, because there is no such frame.

The comparison has already been made in assumption one. It's not logical to say in the next statement the opposite, and say that no we are just not comparing with it's own reference frames. It's a contradiction.

Are Physics textbooks not accurate enough? or have I made a mistake somewhere in my thinking.

I don't really want to be correct, but just want to know my mistake.

But, so far, it seems like a Philosophical mistake in SR.

Any statement is only valid under the assumption applied on top of it.

I understand that the photon cannot have a frame of reference, but what I am pointing to, is that a comparison with "All" reference frames has already been made earlier. Before this statement is established that "Photon cannot have a frame of reference".

You cannot compare without defining. This seems like a paradoxical set of statements.

If you look at the current answers, this is exactly what I am referring to. We have a reason from statement 1 itself for statement 2 to be true, but if statement 2 is true, then it violates part of statement 1, because both cannot be absolutely true together.

I agree that statement 2 will follow from statement 1, but if statement 2 follows, then statement 1 cannot be absolutely true, because it was stated first. Statement 2 follows from it through reasoning.

Let me illustrate what I think could be wrong:
We make statement 1: "Speed of light is same wrt to all reference frames"
These statements are a subset of statement 1:
Statement A: " Speed of light is same wrt to A reference frame"
Statement B: " Speed of light is same wrt to B reference frame"
Statement C: " Speed of light is same wrt to it's own reference frame".

If 1 is true, then A, B , and C all are true, so far.

But, now we say that it follows from 1, that C cannot be true, because there can be no reference frame for photon. This constitutes a contradiction, because C has already been said to be True.

Answer 1

What is the reference frame of an object? It is the frame in which that object is at rest (i.e. it has a speed of zero). If you take the statement

"Speed of light is the same with respect to all reference frames"

to be true, then of course there is no reference frame for the photon: since the speed is the same in all reference frames, there is no reference frame in which the speed is zero.

Regarding your edit: A, B, and C are true assuming those reference frames exist. By making statement C, you are implicitly assuming that the reference frame of the photon exists, and deriving a contradiction from that.

Answer 2

One should be aware that there exist logical frames of reference in addition to the mathematical ones used in studying physics. These are a meta-level on the model's level. Example of meta-levels: the alphabet is one level, the words written with the alphabet is another level, the meaning carried by the words is a third level. Whenever paradoxes appear it is because one mixes levels of reference.

So, I know that SR makes two statements. One is the first assumption it has: "Speed of light is the same with respect to all reference frames".

Second, the statement: "A photon cannot have a reference frame". (Please correct me if the statements I wrote are not accurate.)

At the level of your two statements , SR makes many more statements:

"inertial mass increases with velocity"

"time is different in different reference frames"

Depending on the variables used similar statements can appear in the set of statements. All the statements depend on the underlying mathematical level of SR, and may be similar or connected through it.

Aren't these two statements a bit paradoxical?

All statements of special relativity can seem paradoxical because our everyday intuitions are dependent on classical Galilean physics. But the set of statements comes from Lorenz transformations and is a meta level to the transformations which have as an axiom the uniqueness of the speed of light.

The second statement is mathematically derived using Lorentz transformations and belongs to the set of statements, not to the mathematics of Lorenz transformations.

I agree that statement 2 will follow from statement 1, but if statement 2 follows, then statement 1 cannot be absolutely true, because it was stated first. Statement 2 follows from it through reasoning.

To recapitulate: statement 2 belongs to the set of statements derivable with the mathematics of Lorenz transformations. Statement 1 is primarily an axiom for defining the mathematics of the Lorenz transformations, and appears in the statements level as a redundancy.

Edit after edit of question:

The statements set is derived and stated by using the mathematics of the mathematical level. (the way a word is written by using the alphabet) In the mathematical level itself there exist singularities. In the link for Lorentz transformations one observes that when the velocity of a particle is c an infinity appears:

Statement A: " Speed of light is same wrt to A reference frame" Statement B: " Speed of light is same wrt to B reference frame" Statement C: " Speed of light is same wrt to it's own reference frame".

So your statement C does not belong to the set of statements derivable from the mathematics, because an infinity appears and C is undefined, so cannot have a reference frame. (that is the origin of the statement) You derive it from statements A and B by logic , mixing a higher level with a mathematical under-level. Logic is not enough to do the mathematics of Lorenz transformations.

Answer 3

Time loses its meaning in photon's frame of reference. Past, present, future all happen at the same time according to a photon.

Let's consider why: Speed of light should be same in all reference frames. And all observers see light moving in their own frame of reference. But for the photon, the light would be at rest. There is no frame where light is at rest in the special theory of relativity.