τὰ στοιχεῖα τοῦ κόσμου, the elements (fundamentals?) of the world, that is, any of the world’s rules to live by (or rules for novices/children/basic training), of which Jewish Torah, or a parent’s or guardian’s rules, are examples? (See also στοιχέω, to live by [or be under?] the rules.)
Gal 4:3: Οὕτως καὶ ἡμεῖς, ὅτε ἦμεν νήπιοι, ὑπὸ τὰ στοιχεῖα τοῦ κόσμου { ἦμεν δεδουλωμένοι ♦ ἤμεθα δεδουλωμένοι }· [So also we, when we were minors, were enslaved under the basic forces of the world. (NET)]
Gal 4:9: νῦν δέ, γνόντες θεόν, μᾶλλον δὲ γνωσθέντες ὑπὸ θεοῦ, πῶς ἐπιστρέφετε πάλιν ἐπὶ τὰ ἀσθενῆ καὶ πτωχὰ στοιχεῖα, οἷς πάλιν ἄνωθεν δουλεύειν θέλετε; [But now that you have come to know God (or rather to be known by God), how can you turn back again to the weak and worthless basic forces? Do you want to be enslaved to them all over again? (NET)]
Or “basic principles,” “elemental things,” or “elemental spirits.” Some interpreters take this as a reference to supernatural powers who controlled nature and/or human fate.
Colossians 2:8: Βλέπετε μή τις ὑμᾶς ἔσται ὁ συλαγωγῶν διὰ τῆς φιλοσοφίας καὶ κενῆς ἀπάτης, κατὰ τὴν παράδοσιν τῶν ἀνθρώπων, κατὰ τὰ στοιχεῖα τοῦ κόσμου, καὶ οὐ κατὰ χριστόν· [Be careful not to allow anyone to captivate you through an empty, deceitful philosophy that is according to human traditions and the elemental spirits of the world, and not according to Christ. (NET)]
Colossians 2:20: Εἰ ἀπεθάνετε σὺν χριστῷ, ἀπὸ τῶν στοιχείων τοῦ κόσμου, τί ὡς ζῶντες ἐν κόσμῳ δογματίζεσθε, [If you have died with Christ to the elemental spirits of the world, why do you submit to them as though you lived in the world? (NET)]
The NET Bible notes (http://www.bible.org/netbible/index.htm?col2.htm):
The phrase κατὰ τὰ στοιχεῖα τοῦ κόσμου (kata ta stoiceia tou kosmou) is difficult to translate because of problems surrounding the precise meaning of στοιχεῖα in this context. Originally it referred to the letters of the alphabet, with the idea at its root of things in a row.... M. J. Harris...outlines three probable options: (1) the material elements which comprise the physical world; (2) the elementary teachings of the world (so NEB, NASB, NIV); (3) the elemental spirits of the world (so NEB, RSV). The first option is highly unlikely because Paul is not concerned here with the physical elements, e.g., carbon or nitrogen. The last two options are both possible. Though the Gnostic-like heresy at Colossae would undoubtedly have been regarded by Paul as an elementary teaching at best, because the idea of spirits played such a role in Gnostic thought, he may very well have had in mind elemental spirits that operated in the world or controlled the world (i.e., under God’s authority and permission).τὰ στοιχεῖα τῆς ἀρχῆς τῶν λογίων τοῦ θεοῦ, the elements of the beginning of the oracles of God
Hebrews 5:12: Καὶ γὰρ ὀφείλοντες εἶναι διδάσκαλοι διὰ τὸν χρόνον, πάλιν χρείαν ἔχετε τοῦ διδάσκειν ὑμᾶς, { τίνα ♦ τινὰ } τὰ στοιχεῖα τῆς ἀρχῆς τῶν λογίων τοῦ θεοῦ· καὶ γεγόνατε χρείαν ἔχοντες γάλακτος, { καὶ οὐ ♦ [καὶ] οὐ } στερεᾶς τροφῆς. [For though you should in fact be teachers by this time, you need someone to teach you the beginning elements of God’s utterances. You have gone back to needing milk, not solid food. NET)]
στοιχεῖα (of the universe?)
2 Peter 3:10: Ἥξει δὲ { ἡ ἡμέρα ♦ ἡμέρα } κυρίου ὡς κλέπτης { ἐν νυκτί ♦ - }, ἐν ᾗ οἱ οὐρανοὶ ῥοιζηδὸν παρελεύσονται, στοιχεῖα δὲ καυσούμενα { λυθήσονται ♦ λυθήσεται }, καὶ γῆ καὶ τὰ ἐν αὐτῇ ἔργα { κατακαήσεται ♦ εὑρεθήσεται }. [But the day of the Lord will come like a thief; when it comes, the heavens will disappear with a horrific noise, and the celestial bodies will melt away in a blaze, and the earth and every deed done on it will be laid bare. (NET)]
2 Peter 3:12: προσδοκῶντας καὶ σπεύδοντας τὴν παρουσίαν τῆς τοῦ θεοῦ ἡμέρας, δι’ ἣν οὐρανοὶ πυρούμενοι λυθήσονται, καὶ στοιχεῖα καυσούμενα τήκεται; [while waiting for and hastening the coming of the day of God? Because of this day, the heavens will be burned up and dissolve, and the celestial bodies will melt away in a blaze! (NET)]
The NET Bible notes (http://www.bible.org/netbible/index.htm?2pe3.htm):
Grk elements. Most commentators are agreed that celestial bodies is meant, in light of this well-worn usage of στοιχεῖα (stoiceia) in the 2nd century and the probable allusion to Isa 34:4 (text of Vaticanus).Extra-biblical usages:
Classical Elements (fire, earth, air, water): http://en.wikipedia.org/wiki/Classical_element
Elements of Euclid: http://en.wikipedia.org/wiki/Euclid\\\'s_Elements