Warning: The following is a bunch of philosophical mumbo jumbo.

The Paradox of Identity isn't a paradox. It's a series of competing terminologies used to define an abstract idea.

Identity is based on reference points. In an extreme sense, something can only be identified as being different from something else if there is a reference point from which the one thing differs from the other. This is implicit in the very concept of identity. To say that identity has no formal definition is merely to invite a paradox into your own personal reality.

Example: in a spatial reference frame (room of some sort) there exists a green 5kg rubber cube located at (2,2,0)meters. If the "green 5kg rubber cube located at (2,2,0)meters" is moved to (2,1,0), the subject of this example is still a "green 5kg rubber cube". If the "green 5kg rubber cube" subject is deformed into a sphere, the subject is still green, 5kg in mass, and composed of rubber. This process of systematically changing only one property of the subject could be repeated until the subject shared no properties with its initial self. The subject is now a blue 50kg steel sphere located at (2,1,0)meters. The question posed by philosophers is this: does the current subject have the same identity as the initial subject?

My answer to this: it depends on the reference frame. It seems obvious that, since the subject of the discussion has not changed to another object, the current subject must share identities with the initial subject. And, it is also obvious that I can't reference the current subject without first having some frame of reference. But this is getting confusing, so let's assign names. The "green 5kg rubber cube located at (2,2,0)meters" is called Smith. The "blue 50kg steel sphere located at (2,1,0)meters" is called Jones. The question now becomes: is Smith identical to Jones? Do these subjects share an identity? Now, I'd have to say no, they don't. They are in every way discernible from one another. So, because they share no transitive quality, they are in no way identical. They are different. They are not the same. Actually, there is one transitive quality: Smith becomes Jones. We know this is true because the subject of the sentence has always been the previous subject with 1 change in property. But, if the subject wasn't the subject I would be talking about nothing... So, I can't talk about a thing with out talking about it and thereby providing a reference frame... This is becoming too abstract to consider. We need something concrete to discuss.

All of this rambling is mere nonsense if there is no practical application, so I shall provide one: You. What is your identity? Look at a picture of yourself from 5 minutes ago/5 days ago/5 years ago. Are you the same person? Define the following word: I.

This universe in which we live is entirely relational. "That the universe is relational" is a continuous property of the universe. Because it doesn't change, I can define the universe. 

Living in this universe consists of comparisons of states from reference frames. These comparisons tend to fluctuate as time goes on and nature "works its course". Our bodies are constantly changing. Therefore, they can act only as an instantaneous identifier. Our thoughts are also constantly changing. What can be done to identify one person from another?

I have a proposition. We should give names to each other. We could also identify persons from one another by cause and effect. When two people are responsible (physically speaking) for the creation of a third, we will call the responsible male "Father" and the female "Mother". The created third will be called the "Child" or "Offspring" of the father and mother. These qualities will never change and can therefore be used as an unchanging, constant identifier. That a child's mother and father continue to be his/her mother and father is constant. The relation persists. It does not change or fluctuate. 

The father and mother should issue the child his/her name. It would be fitting. Obviously, the child will be too young when it is born to even grasp the idea of a name. So, it wouldn't be prudent to name it "goobleeburwaaaa" or whatever we thought it was saying. Now, this child will be forever identified as [Name], [son/daughter] of [father] and [mother]. We should record all of this on a document and include with it a frame of reference for the child. By having this frame of reference the child will be able to live relationally and sensibly (or so we hope). It can be arbitrary, but it needs to be a thing that happens only once and never again or else things will get confusing for the child... How about the date? Yes, that should work. We will issue this child a "Date of Birth" from which he/she can calculate the number of days he/she has lived among human beings. Then, he/she can make relational comparisons and have an identity. The world can make sense to the child.

This lengthy situational description only serves to show that, indeed, the universe is relational. It's implicitly understood that when a person "A" says for another "B" to "define" a word "shroobiness", A wants B to put shroobiness into terms that are somehow constant but are also capable of being referenced. So, when A says, "Shroobiness is that feeling you get after sitting in a chair (or other sort of furniture) that is already warm from someone else's having just previously sat in it." B can now perfectly comprehend shroobiness. A now knows that he feels shrooby. A knows what it means to feel shrooby. Shroobiness depends on relative temperatures, furniture of a sort (relative shapes of solid materials), time (ralative to when the other person left the chair), and relative posture. The word has been defined. It has been described using relational concepts.

If this seems false, define something that depends on nothing else's point of reference.  Then put it in the comments.

The Paradox of Identity:

• Some things persist.
• If thing "X" and thing "Y" are identical, then every property X has is one that Y has also. (This is the "Indiscernibility of Identicals")
• If something persists, then it has an age at one time and another age at another time.
  

The paradox applies to self identity. I.E. looking at a picture of your past self and saying "That's me". According to these premises, "that" is not "me"...well, you in this case. Present You is older than Past You. So, you aren't identical with your past self. If you're not identical, you share no identity... Are you the same person? Are you still yourself? You have persisted. You have had different ages at different times. Then, is your identity the same? It appears as though the first premise must be false. But, it also appears as though you still exist...paradox.

As was stated earlier though, The Paradox of Identity isn't a paradox. It's a series of competing terminologies used to define an idea. According to the second premise, X and Y are identical only if there is no difference between them. Identity is thereby turned into a binary sort of quality. Either they are identical or they aren't. But, if "they" are identical, "they" must occupy the same space so they have the same spatial property. So, "they" are really just "it". There can be no 2 identical things. Identical twins don't exist. If you previously knew any identical twins, say a prayer for them...they no longer exist. That's the essence of Premise 2.

Premise three says that "If something persists, then it has one age at one time and another at another time". Wait a minute, if that's true and if "Some things persist", then persistence must allow for a change in age (time property). But, the thing that persists is then not identical to its previous self. It loses its identity in time. Now, it's back to denying the truth of the first premise.

The problem in the premises is the lack of an outer reference frame. If a definition is truly a relational quality, and our minds really only understand concepts or ideas as they relate to other concepts or ideas, then the "Paradox" quality of the argument disappears when we give the "persisting things" names. We could give them any constant identifier to bring end to the paradox. Let me explain further.

John persists in that he continues to be John and to carry the qualities of himself. When those qualities change, he is still John. His identity remains intact through change because he has a reference frame in which nothing changes. In his name-reference frame, he has a name: John. That never changes, so he always has an identity. However, his identity does change. One could visualize an identity as an infinitely long check list of relational qualities that one either does or doesn't have. If identity can be viewed that way, a person can have a changing identity. But, his current identity is never identical to his past identity.

• John persists.
• If "20 year-old John" and "40 year-old John" are identical, then every property "20 year-old John" has is one that "40 year-old John" has also. (This is the "Indiscernibility of Identicals")
• If John persists, then he has an age at one time and another age at another time.
  

Now, these premises are all true. Simply put, "20 year-old John" is not identical to "40 year-old John". So there is no paradox at all.