Jesus Christ the Son of God
And truly Jesus did many other [miraculous] signs in the presence of His disciples, which are not written in this book; but these are written that you may [continue] believe that Jesus is the Christ, the Son of God, and that believing you may have life in His name.
Taking the book as a whole, these two verses form a parenthesis, which could be paraphrased as:
"I have recorded in this gospel a few of the signs that I and the other disciples saw Jesus perform: there are many more, some of which are recorded in the other Gospels, however, I have expanded upon eight signs (seven before and one after the resurrection), along with proof of his resurrection, giving you abundant evidence that Jesus is the Son of God, the Messiah, and that if you believing in Him, you will have everlasting life."
Or taking these verses as a follow-on from the resurrection, not as a parenthetical paragraph, but a continuation of chapter 20:
"As I have just explained, Jesus rose from the dead and with many proofs demonstrated this to be true, of which I have explained just some of them to you: there were three appearances to us. He did many other signs proving his resurrection, but I have not recorded these in my gospel. The ones I have recorded, however, demonstrate adequately that Jesus is the Son of God, and if you believing in the resurrected Son of God, you will have everlasting life."
What do these verses say
1. The subject of the book which John wrote is Jesus Christ. The author here points out that there are other matters pertaining to Christ's resurrection that could have been included, but he, under the guidance of the Holy Spirit, selected a few and these are sufficient enough as proof of the deity of Christ.
2. These verses follow-on from John's explanation of the resurrection of Jesus. Clearly Jesus appeared to them many times and probably performed many other signs, as these two verses indicate. However, John explained the resurrection given describing seven signs:
a) The stone taken away from the tomb's mouth
b) The empty tomb
c) The manner in which the garments used to wrap Jesus were lain
d) The presence of two angels in the tomb
e) The first appearance of Jesus to Mary Magdalene
f) The second appearance of Jesus to his disciples in the closed room
g) The third appearance in like manner in a closed room
3. The proof of the argument that Jesus was the Messiah is that he rose from the dead and appeared at least three times: two if not three miraculously (see also Romans 1:4 and Acts 13:33 that quotes Psalm 2). Jesus came into the world professing to be the Son of God: his resurrection proved it, because it established the truth of everything he taught.
4. Jesus had disciples, that is, followers of himself. These were eyewitness to the events that befell Jesus, hence are our prime information source. It is clear the Jesus showed many signs to his disciples and others, at least 500 after his resurrection (1 Corinthians 2:15) that proved that the man standing before them was Jesus, the man they knew before his death:
Him God raised up on the third day, and showed Him openly, not to all the people, but to witnesses chosen before by God, even to us who ate and drank with Him after He arose from the dead. Acts 10:40-41
5. The signs were to the disciples so they could demonstrate the truth to the world:
That which was from the beginning, which we have heard, which we have seen with our eyes, which we have looked upon, and our hands have handled, concerning the Word of life-1 John 1:1 (NKJV)
6. The purpose of the book of John is that we may believe, or continue to believe, whatever our position, that Jesus, the subject of the book, is Christ, the Son of God and to show that Christ came to reveal the Father to us .
7. The outcome of this belief is a blessing beyond all measure: life in the name of Jesus.
The second paraphrase overcomes the difficulties in explaining a parenthesis that deals with the whole book, rather than the previous chapter.
The Object of the Verses
Jesus Christ, the incarnate God, the Messiah, the Son of God, proved by His resurrection.
See 1 John 5:9
10He who believes in the Son of God has the witness in himself; he who does not believe God has made Him a liar, because he has not believed the testimony that God has given of His Son. 1 John 5:10 (NKJV)
The subject of the Verses
The subject of the verses is encouragement that we have in knowing Jesus was the Messiah. Thus John invites us to believe in Jesus Christ: for without faith it is impossible to please God, for we cannot be saved without faith.
The Son of God
First mentioned in Daniel - in fear King Nebuchadnezzar looked into the fiery furnace and saw four men, one of which had the appearance of the Son of God (Daniel 3:25).
Jesus declared to be the Son of God (Romans 1:4).
Jesus, the Messiah, was promised through the prophets, and was born of a woman, as the seed of David. This promise arose out of the sin of Adam: Genesis 3:15.
And I will put enmity Between you and the woman, And between your seed and her Seed; He shall bruise your head, And you shall bruise His heel." Genesis 3:15 (NKJV)
By one man sin entered the world.
Sin brought death to mankind, for the wages of sin is death (Rom 6:23), and the only remedy was for God to provide His son.
John writes of this in 1 John 3:7 - the reason the Son of Man was made manifest, that is, made known to us, was to deal with the Devil and to destroy his works:
Inasmuch then as the children have partaken of flesh and blood, He Himself likewise shared in the same, that through death He might destroy him who had the power of death, that is, the devil, Hebrews 2:14 (NKJV)
The declaration of the book of Romans is exactly the same as seen in John 20:30,31. The declaration is by the resurrection of Jesus, though the Holy Spirit.
Greater than Moses
This same Jesus, declared to be the Son of God, is also revealed as the Great High Priest (Hebrews 4:14). A little before in Hebrews He is shown to be greater then Moses in that: although Moses was faithful in all of God's house, as God himself declared in Numbers 12:7, this Jesus was actually the builder of the House, and not only that, He rules over it, thus demonstrating his superiority to Moses.
Our Great High Priest
Moses was not a High Priest, yet Jesus is a high priest, and much more, a high priest, which is superior to Aaron, for Aaron could not take away the sin of the people. Noting also that Aaron was not a king, and Jesus is a King and High Priest, after the order of Melchizedek, not Aaron. What does this mean?
Hebrews 7 explains what the order actually was - it was one without beginning or ending: hence Jesus, the alpha and omega, is not forever, and hence our priest for ever, continually (Hebrews 7:3) interceding on our behalf before the Father.
The Father sees us only through Jesus, the blood cleansing us from all unrighteousness. There is nothing in us that allows the Most Holy One to look upon us, only the work Christ was wrought does remove this barrier.
Eyes like a flame of fire, feet like fine brass (Rev 2:18)
These characteristics highlight the Son of God's ability to search all - there is nothing hid from him - his eyes search all, and sees. All sin needs to be dealt with - holiness means that there is no sin, no not one. The Son of Man is able to deal with everyone of them:
"For His eyes are on the ways of man, And He sees all his steps. Job 34:21 NKJV)
What is hidden he brings forth to light. Job 28:11 (NKJV)
For God will bring every work into judgment, Including every secret thing, Whether good or evil. Ecclesiastes 12:14 (NKJV)
His feet are like fine brass speaking of his holy walk which stands the testing of God, as the consuming fire. (Deut 4:24).
The Son of God came for the express purpose to deal with sin. He did not come to condemn the world, but to save it, from its iniquity. Because we were aliens, afar off, indeed an unpassable gulf existed between God and man prior to this that none could conquer. Jesus provided a means to reconcile us to God - Ephesians 2:16.
Truth and grace.
He reveals the Father which Truth and Grace. Christ is the Truth. By grace we are saved.
The Son of God
He has re-entered heaven as Man, to be head over all things.
Has been appointed heir (Hebrews 1).
He has been the manifestation of God upon this earth.
He will reign over the earth as the Head of all creation (of all things), gathering together all things under His authority (Colossians 1).
All things are to be subjected under His feet - He, entering heaven as Man, is in the presence of God without a veil.
He redeems those that are to enter the Father's presence.
The most amazing thing here is Christ enters heaven as a Man, who God can look upon.
 These are brief notes used for a short meditation on the given passage. By no means are they an in-depth discussion on the fullness of the words of Jesus. (Bible Truth). Furthermore, the author expects that the grammar could be somewhat better represented.
 The signs Jesus did in John
2:1 Turning water into wine
4:46 Healing the Nobleman's son of fever
5:1 Healing an infirm man at Bethesda
6:1 Feeding the Five Thousand (also M,M & L)
6:19 Walking on water (also M & M)
9:1 Healing the man born blind
11:43 Raising of Lazarus
21:1 Catch of fish - after disciples had failed to catch any.
 This presupposes that there is a God, and furthermore, He has a Son that come to this world as a man, that walked with the disciples.