I think this could be a somewhat interesting topic. Obviously it will help me practice my own presentations on philosophy, so i might as well create it here 
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The first philosopher, according to Aristotle, was Thales of Miletus, in the mid 7th century BC. However the second in that school, his student Anaximander, was the first to speak of a notion of a 'limitless', or 'infinite' (although the meaning of infinite was secondary to his idea, primarily it was the meaning of something without any limit, or even form). 'Apeiron', the term still used in math here, or else for the idea of infinity or no limit, was Anaximander's primal or primordial element for the cosmos- or rather in plural, the cosmoi, cause he argued that the worlds (either star systems, or even whole universes, by current terminology) were many and not just one.
1) Anaximander, Democritus and Protagoras
Anaximander lived in the early 6th century BC. His own student was Anaximenes, the third of the Milesian philosophers, and possibly also Pythagoras of Samos. While Pythagoras later on moved to the other edge of the Greek world, to Italy and the colonies there, Anaximenes remained a part of the Milesian (and overall Ionian) school of philosophy, and in many respects can be said to have been closer to Thales than Anaximander. For example Anaximenes also defined a prime building substance of the cosmos, as Thales had. Thales proposed Water as that substance. Anaximenes proposed Air, arguing that the changes in density lead it to at one end become fire (less density) and the other end earth (more density). He is mostly argued to have been a father of the notion of examining opposites in physical circumstances, such as heat or density.
Democritus lived a century after Anaximenes. He was a defining philosopher of the school of Abdera, a Greek colony in coastal Thrace. His own main theory was that any material form breaks up to smaller forms, which he termed as "Atoma", ie atoms, the term used later on by modern physics. Democritus, such as the other notable philosophers linked to Abdera (like Protagoras, a former servant who was took under the protection of Democritus and became his student), can be said (in my view) to be (as geographically as well) a middle point or a synthesis between the ends of the Ionian and the Italiotic Greek philosophical movements. At least by and large.
Protagoras arrived in Athens following the defeat of the Persians in their second and final expedition against mainland Greece. He was the most notable of the sophists, and argued also to have been an originator of their order. The Sophists were paid teachers of ways of thinking in a number of subjects, from math to political thought, to more abstract philosophy. They are presented in the Platonic dialogues, speaking with Socrates.
2. The Infinite of Anaximander as a beginning before the current progression, and as an ending after the current progression.
Anaximander's most famous saying (of those which have been saved in some form) is quoted by later, mostly early Byzantine, scholars, such as Simplicius in the 6th century AD. According to it: "The origin (Arche) of the cosmoi is the limitless/indefinate/infinite, from which they came to be, and to which they return to be destroyed back to, according to the order which seeks them to pay back what was given to them, the order of Time (Chronos)".
This saying was examined by various thinkers in later millenia too, such as Nietzsche, who sees in it a sort of punishing quality in the current cosmos. But the Apeiron of Anaximander was primarily something outside (by definition, as an axiom) of what we can study. An origin of the cosmoi which was axiomatically set outside of the grasp of humans. This abstraction plays a very important role in anything which followed in Greek philosophy.
3. Protagoras and Anaxagoras in Athens, 5th century BC
Anaxagoras was from Ionia, the city of Klazomenae, which along with Ephesos, Miletos and Kolophon were the main Greek cities in Asia linked to philosophy in those periods. His own particular contribution was to argue that the borderless which Anaximander spoke of 2 centuries before, was borderless to one direction, but not another. Ie it was the human thinking, the human thinker, who could examine anything outside his own conscious thought, in a more expanded manner, while in a more problematic manner to also look to his own conscious thought and analyse it to equal complexity. In other words Anaxagoras argued that it was Logic (which in his work also means the consciousness, or its center) that gave form to anything one could observe, and moreover itself could not be examined in the same manner, cause it could not look to its own self with the same freedom. Self-introspection, of course, is a different order of examination than examining external objects or phenomena.
Protagoras argued famously that "Man is the meter of all things", which seems to me to add another stability to the border set by Anaxagoras, and before him Anaximander. Now not only is consciousness the stable point below which things can blurred and one can hardly seek an origin/arche, but it is the specific human species which sets the border through largely non-conscious as to their reason, similarities. For example we may have very different views in a myriad of things, but all humans will see a triangle as a triangle and not as a circle (provided they know how each shape is called).
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Well, this was a sort of brief synopsis of the arguments for the central idea of that which is without limit, in ancient Greek Philosophy. Expanded it includes the Eleatic philosophers (Parmenides, Zeno, to an extent also Pythagoras) and the position that the Origin/Arche is by itself not just axiomatically distant and out of reach, but non-existent. The synthesis of the two views, the Ionian of a defined Origin, the Eleatic of a non-defined origin, would be the view presented above, of a clear border, to sides of which lay unrecongnisable, obscure, shadowy origins, which serve as a limit to philosophy.
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Thank you if you took the time to read all that... RD'd it just cause it became a bit larger as an OP than i originally meant, and would not feel that well if met with some comment outside of the topic
*
The first philosopher, according to Aristotle, was Thales of Miletus, in the mid 7th century BC. However the second in that school, his student Anaximander, was the first to speak of a notion of a 'limitless', or 'infinite' (although the meaning of infinite was secondary to his idea, primarily it was the meaning of something without any limit, or even form). 'Apeiron', the term still used in math here, or else for the idea of infinity or no limit, was Anaximander's primal or primordial element for the cosmos- or rather in plural, the cosmoi, cause he argued that the worlds (either star systems, or even whole universes, by current terminology) were many and not just one.
1) Anaximander, Democritus and Protagoras
Anaximander lived in the early 6th century BC. His own student was Anaximenes, the third of the Milesian philosophers, and possibly also Pythagoras of Samos. While Pythagoras later on moved to the other edge of the Greek world, to Italy and the colonies there, Anaximenes remained a part of the Milesian (and overall Ionian) school of philosophy, and in many respects can be said to have been closer to Thales than Anaximander. For example Anaximenes also defined a prime building substance of the cosmos, as Thales had. Thales proposed Water as that substance. Anaximenes proposed Air, arguing that the changes in density lead it to at one end become fire (less density) and the other end earth (more density). He is mostly argued to have been a father of the notion of examining opposites in physical circumstances, such as heat or density.
Democritus lived a century after Anaximenes. He was a defining philosopher of the school of Abdera, a Greek colony in coastal Thrace. His own main theory was that any material form breaks up to smaller forms, which he termed as "Atoma", ie atoms, the term used later on by modern physics. Democritus, such as the other notable philosophers linked to Abdera (like Protagoras, a former servant who was took under the protection of Democritus and became his student), can be said (in my view) to be (as geographically as well) a middle point or a synthesis between the ends of the Ionian and the Italiotic Greek philosophical movements. At least by and large.
Protagoras arrived in Athens following the defeat of the Persians in their second and final expedition against mainland Greece. He was the most notable of the sophists, and argued also to have been an originator of their order. The Sophists were paid teachers of ways of thinking in a number of subjects, from math to political thought, to more abstract philosophy. They are presented in the Platonic dialogues, speaking with Socrates.
2. The Infinite of Anaximander as a beginning before the current progression, and as an ending after the current progression.
Anaximander's most famous saying (of those which have been saved in some form) is quoted by later, mostly early Byzantine, scholars, such as Simplicius in the 6th century AD. According to it: "The origin (Arche) of the cosmoi is the limitless/indefinate/infinite, from which they came to be, and to which they return to be destroyed back to, according to the order which seeks them to pay back what was given to them, the order of Time (Chronos)".
This saying was examined by various thinkers in later millenia too, such as Nietzsche, who sees in it a sort of punishing quality in the current cosmos. But the Apeiron of Anaximander was primarily something outside (by definition, as an axiom) of what we can study. An origin of the cosmoi which was axiomatically set outside of the grasp of humans. This abstraction plays a very important role in anything which followed in Greek philosophy.
3. Protagoras and Anaxagoras in Athens, 5th century BC
Anaxagoras was from Ionia, the city of Klazomenae, which along with Ephesos, Miletos and Kolophon were the main Greek cities in Asia linked to philosophy in those periods. His own particular contribution was to argue that the borderless which Anaximander spoke of 2 centuries before, was borderless to one direction, but not another. Ie it was the human thinking, the human thinker, who could examine anything outside his own conscious thought, in a more expanded manner, while in a more problematic manner to also look to his own conscious thought and analyse it to equal complexity. In other words Anaxagoras argued that it was Logic (which in his work also means the consciousness, or its center) that gave form to anything one could observe, and moreover itself could not be examined in the same manner, cause it could not look to its own self with the same freedom. Self-introspection, of course, is a different order of examination than examining external objects or phenomena.
Protagoras argued famously that "Man is the meter of all things", which seems to me to add another stability to the border set by Anaxagoras, and before him Anaximander. Now not only is consciousness the stable point below which things can blurred and one can hardly seek an origin/arche, but it is the specific human species which sets the border through largely non-conscious as to their reason, similarities. For example we may have very different views in a myriad of things, but all humans will see a triangle as a triangle and not as a circle (provided they know how each shape is called).
*
Well, this was a sort of brief synopsis of the arguments for the central idea of that which is without limit, in ancient Greek Philosophy. Expanded it includes the Eleatic philosophers (Parmenides, Zeno, to an extent also Pythagoras) and the position that the Origin/Arche is by itself not just axiomatically distant and out of reach, but non-existent. The synthesis of the two views, the Ionian of a defined Origin, the Eleatic of a non-defined origin, would be the view presented above, of a clear border, to sides of which lay unrecongnisable, obscure, shadowy origins, which serve as a limit to philosophy.
---
Thank you if you took the time to read all that... RD'd it just cause it became a bit larger as an OP than i originally meant, and would not feel that well if met with some comment outside of the topic
(well, it just helps me sum up the stuff to present, anyway..).
(but i look better than him; it seems he mostly looked like Tyrion).
post more Kyra when summer holiday finally come, I will devour it for sure. Also you can record your lecture if it is in English, and upload it in youtube, that will be awesome I think.
