If I am infinitely old , can I have a father ? And can I have a brother that is infinitely older than me but younger than my dad ?
A simple model of infinitely deep infinite time
Here's a model for the ordering of events in time such that you can have three different objects, each infinitely old, and each infinitely older than the last. Informally, it involves time not only having an infinite past, but a "very very" infinite past.
We typically represent time by real numbers, possibly extended to include an idealized point in the infinite past ("minus infinity") or infinite future ("plus infinity"). However, we can also concieve of orderings of time in which there isn't just one time in the infinite past, but entire timelines in the infinite past as well.
We might represent time then by a pair of labels: a label E for the epoch, and a label t for the time within the epoch (as measured from some fixed event during that epoch). Most importantly, we must be able to describe an order on the labels of the epochs: for instance, we may label them by numbers. These labels might be drawn from
For the idea you propose, it doesn't really matter which of these we consider, or whether we consider some other model for the number and structure of the epochs, so long as there are at least three epochs — one for the older brother to be born in, one for the younger brother to be born in, and one for the present day. Times are given by ordered pairs (E,t), where (E,t) < (E',t') if either E < E', or E = E' and t < t'. (In the latter case, the difference in time between the events is t' − t; but in the former case, the first event precedes the second by an infinite duration goverened by the difference between the epochs.)
Hypothetically, the father doesn't have to be "born" in any epoch, unless there are things which are meant to be older than him — even something finitely older than him would imply that there was at least one moment in time in which the father didn't exist, and that moment has to belong to an epoch under this model of time. So, just having three epochs would suffice, although there could be more.
On causality in infinite stretches of time
As to cause and effect, there is a reasonable question as to how an event in one epoch affects events infinitely further in time. The very existence of these infinitely old creatures across the epochs is one example of that: the father, for instance, may reasonably be construed as causing (among other things) his own continued existence. The notion of time that I've put forward doesn't give any description of what a continuous sort of causality would look like which ties together the epochs; perhaps if these infinitely aged creatures went into a sort of dormant state, and then awoke in another epcoh, you could simply define the continuity of their existence in terms of convergent states of their behaviour agreeing with each other going into the infinite future of one epoch and the infinite past of another. This implicitly invokes a notion of topology both on the timeline stretching across epochs, but also on the behaviours of the creatures. For instance, we probably demand some serious stability of these infinitely aged creatures if we want to prevent the argument that the creatures in one epoch were swapped out with a different set of infinitely aged creatures in another epoch.
As to how to treat causality of these infinitely aged creatures on their presumably less permanent environment, it's hard to say. If the infinitely old creatures are the only fixed points of the world, so that everything about them is subject to change, then it is only meaningful if we suppose that there are meaningful features of the world as a whole in one epoch which arose as a result of the behaviour of the old ones in a previous epoch.
But these examples are still of infinitely enduring features of the world: the existence of an enduring phenomenon/species in one, and an enduring object in another. Even if we supposed that these infinitely aged creatures could create or destroy energy, the level of energy would be in a sense an enduring excitation of the matter/light fields caused by these creatures akin on a subtler level to making an enduring artefact (which itself is also just a very stable excitation of matter fields). Perhaps the only possible sense in which you can define continuity is in such infinitely enduring things, including the creatures themselves.
Let me formalize your question and see where it gets us.
Let's view time as a set of instants. In order for your question to make sense, this set must have an order (a notion of "comes before") and be infinite. Let us additionally take the standard assumption that time has a minimal element (the big bang) but no maximal element.
Call the minimal element 0 and some other element 1. My father could be born at time 0 and I at time 1 and we both live forever. This doesn't seem to violate our intuitions of causality, and we have a father and child who are both infinitely old.
You can modify this set up (what if time has a maximal element but no minimum? Neither? Both? Is time discrete?) or you can define "older" in a different way (must the sets be capable of being put into bijection? Should we care about the ratios of their ages or their absolute differences?). Depending on how you define such things, the answer might change.
i feel like this is almost more of a linguistic question. the question becomes much simpler if you have a clear understanding of what the word "infinite" means. the word basically means without limits or bounds. if you think of time as linear, and you have a birthday, but will never die for all of eternity, then you are not infinitely old. you can only be infinitely old if you were never born and always existed.
infinity isn't just endless, it's beginingless as well.
Being infinitely old means you've got to mean slightly different relations by the terms "father", "son", "brother" etc. Something along these lines happens in the beginning of Silmarillion, when Illuvatar (~God) creates some ainur (~angels) as brothers and sisters. They aren't born, but are created already in a sibling relationship. Infinitely old beings can have always existed in their particular network of familiar relationships:
An infinitely old creature can always have had an infinitely old father. A father is more than just a creature that have begat you, he also brings you up, helps you learn your way around the universe, teaches you what it means to be a male in this world etc. In the infinitely old family a father is forever and forever fulfilling this role. Such a father didn't cause you to start existing, but have always been shaping you to be as you are.
Same thing with the infinitely "older" brother - in the infinitely old family his being "older" isn't defined in a temporal sense, but in hierarchical. He helps you with details of living father has no patience with, maybe bullies you lovingly, but then stands up for you when other creatures attack.
You say this thing is older than the other, when both has measurable and finite age. But, in case of the infinite thing, it's neither measurable nor finite, which leads => you are using terms which are not applicable to the situation.
Like in Mathematics, you say (45+9 i ) is a complex number, it has its own properties, but elementary, Sine-function isn't applicable to it (like are other functions, i.g. Cosine, Exponential... etc).
Also, there can't be more than one infinitely aged existence simultaneously, as each infinite existence would acquires the whole time and leave nothing to the other one.