Aorist and present subjunctives in John 10:38

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John 10:38 New International Version
But if I do them, even though you do not believe me, believe the works, that you may know and understand that the Father is in me, and I in the Father.”
Nestle Greek New Testament:
38b ἵνα γνῶτε καὶ γινώσκητε
ἵνα (hina)
Conjunction
Strong's 2443: In order that, so that. Probably from the same as the former part of heautou; in order that.

The conjunction hina signifies the beginning of a hina subjunctive purpose clause: so that, in order that ..., The idea is that doing X in order that Y would happen. In this case, Jesus had performed miraculous works, in order that (for the purpose that) ... .

What is Y?

In this case, there are two items: γνῶτε καὶ γινώσκητε (NIV know and understand).

you may know
γνῶτε (gnōte)
Verb - Aorist Subjunctive Active - 2nd Person Plural Strong's 1097: A prolonged form of a primary verb; to 'know' in a great variety of applications and with many implications.

understand
γινώσκητε (ginōskēte)
Verb - Present Subjunctive Active - 2nd Person Plural
Strong's 1097: A prolonged form of a primary verb; to 'know' in a great variety of applications and with many implications.

These are two forms of the same Greek verb γινώσκω (G1097) to know, as in Adam knew Eve.

The subjunctive is rendered as "may".

What is the difference between aorist and present?

The aorist refers to the action without saying anything about its continuance or repetition (could be a single punctiliar action), while the present refers to it as continuing or as being repeated.

Cambridge Bible for Schools and Colleges:
that ye may know, and believe] The better reading probably is, that ye may come to know and continually know; ‘attain to knowledge and advance in knowledge in contrast to your state of suspense’ (John 10:24). In the Greek it is the aorist and present of the same verb ‘to come to know, perceive, recognise:’ the aorist denotes the single act, the present the permanent growth. The apparent awkwardness of having the same verb twice in the same clause has probably caused a large number of authorities to substitute another verb in the second case. But the change of tense is full of meaning, especially in reference to the Jews. Many of them attained to a momentary conviction that He was the Messiah (John 2:23, John 6:14-15, John 7:41, John 8:30, John 10:42, John 11:45); very few of them went beyond a transitory conviction (John 2:24, John 6:66, John 8:31).

Amplified Bible:
But if I am doing them, even if you do not believe Me or have faith in Me, [at least] believe the works [that I do—admit that they are the works of God], so that you may know and keep on knowing [clearly—without any doubt] that the Father is in Me, and I am in the Father [that is, I am One with Him].”

My paraphrase:
But if I do them, even though you do not believe me, believe the works, that you may know once and for all that the Father is in me, and I in the Father.”